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        "scratch": "x=-4, y=6:  \nA: abs((abs((9*(-4)//5 - 3) + 7)//4 - 9) + abs((abs((4*6)//2 - 6) + 3)//5 - 4))  \n\u2192 9*(-4) = -36 //5 = -7 \u2192 abs(-7 -3)=10 \u2192 10+7=17 \u2192 17//4=4 \u2192 4-9=-5 \u2192 abs(-5)=5  \n\u2192 4*6=24 //2=12 \u2192 12-6=6 \u2192 abs(6)=6 \u2192 6+3=9 \u2192 9//5=1 \u2192 1-4=-3 \u2192 abs(-3)=3  \n\u2192 5+3=8 \u2705  \n\nx=-9, y=3:  \nA: abs((abs((9*(-9)//5 -3)+7)//4 -9)) + abs((abs((4*3)//2 -6)+3)//5 -4)  \n\u2192 9*(-9) = -81 //5 = -16 \u2192 abs(-16-3)=19 \u2192 19+7=26 \u2192 26//4=6 \u2192 6-9=-3 \u2192 abs=3  \n\u2192 4*3=12 //2=6 \u2192 6-6=0 \u2192 abs=0 \u2192 0+3=3 \u2192 3//5=0 \u2192 0-4=-4 \u2192 abs=4  \n\u2192 3+4=7 \u274c expected 6 \u2192 fail\n\nB: abs((abs((10*x)//5 -3)+7)//4 -9) + abs((abs((5*y)//2 -6)+3)//5 -4)  \nx=-9: 10*(-9) = -90 //5 = -18 \u2192 -18-3 = -21 \u2192 +7 = -14 \u2192 //4 = -4 \u2192 -4-9 = -13 \u2192 abs=13  \ny=3: 5*3=15 //2 = 7 \u2192 7-6=1 \u2192 +3=4 \u2192 //5=0 \u2192 0-4=-4 \u2192 abs=4 \u2192 13+4=17 \u274c expected 6  \n\nC: same as B \u2192 same failure \u2192 fail  \n\nD: abs((abs((9*x)//5 -3)+7)//4 -9) + abs((abs((5*y)//2 -6)+3)//5 -4)  \nx=-9: 9*(-9) = -81 //5 = -16 \u2192 -16-3 = -19 \u2192 +7 = -12 \u2192 //4 = -3 \u2192 -3-9 = -12 \u2192 abs=12  \ny=3: 5*3=15 //2 = 7 \u2192 7-6=1 \u2192 +3=4 \u2192 //5=0 \u2192 0-4=-4 \u2192 abs=4 \u2192 12+4=16 \u274c expected 6  \n\nSo A failed at x=-9,y=3 \u2192 fails\n\nWait \u2014 earlier A gave 8 for first pair, but failed second.\n\nBut let's recheck A on first pair:\n\nx=-4, y=6  \n(9*x)//5 = (9*(-4))//5 = -36//5 = -8 (since floor division rounds down)  \n-36//5 = -8 \u2192 -8 -3 = -11 \u2192 abs(-11)=11 \u2192 11+7=18 \u2192 18//4=4 (18//4=4.5\u21924) \u2192 4-9=-5 \u2192 abs=5  \n(4*y)//2 = (24)//2=12 \u2192 12-6=6 \u2192 abs=6 \u2192 +3=9 \u2192 9//5=1 \u2192 1-4=-3 \u2192 abs=3  \n5+3=8 \u2705\n\nx=-9, y=3  \n9*(-9) = -81 //5 = -17 (since -81//5 = -16.2 \u2192 -17)  \n-17 -3 = -20 \u2192 abs=20 \u2192 +7=27 \u2192 //4 = 6 (27//4=6) \u2192 6-9=-3 \u2192 abs=3  \n5*3=15 //2 = 7 \u2192 7-6=1 \u2192 +3=4 \u2192 //5=0 \u2192 0-4=-4 \u2192 abs=4 \u2192 3+4=7 \u274c expected 6\n\nSo A fails.\n\nNow check if any expression gives 6 for x=-9,y=3.\n\nTry candidate B again:\n\nx=-9: (10*x)//5 = -90//5 = -18 \u2192 -18 -3 = -21 \u2192 +7 = -14 \u2192 //4 = -4 (since -14//4 = -4) \u2192 -4-9 = -13 \u2192 abs=13  \ny=3: (5*3)//2 = 15//2 = 7 \u2192 7-6=1 \u2192 +3=4 \u2192 //5=0 \u2192 0-4=-4 \u2192 abs=4 \u2192 13+4=17 \u274c\n\nC: same as B \u2192 17 \u274c\n\nD: (9*x)//5 = -81//5 = -17 \u2192 -17-3=-20 \u2192 +7=-13 \u2192 //4 = -4 (since -13//4 = -4) \u2192 -4-9=-13 \u2192 abs=13  \n(5*y)//2 = 15//2=7 \u2192 same \u2192 4 \u2192 abs=4 \u2192 13+4=17 \u274c\n\nAll fail? But one must work.\n\nWait \u2014 perhaps I miscalculated the first pair.\n\nTry candidate A again with x=1, y=8 \u2192 expected 9\n\nA:  \n(9*1)//5 = 9//5 = 1 \u2192 1-3 = -2 \u2192 abs=2 \u2192 +7=9 \u2192 //4 = 2 \u2192 2-9 = -7 \u2192 abs=7  \n(4*8)=32 //2=16 \u2192 16-6=10 \u2192 abs=10 \u2192 +3=13 \u2192 //5=2 \u2192 2-4=-2 \u2192 abs=2  \n7+2=9 \u2705\n\nx=11, y=-6 \u2192 expected 3  \n(9*11)=99//5=19 \u2192 19-3=16 \u2192 abs=16 \u2192 +7=23 \u2192 //4=5 \u2192 5-9=-4 \u2192 abs=4  \n(4*(-6))=-24 //2 = -12 \u2192 -12-6=-18 \u2192 abs=18 \u2192 +3=21 \u2192 //5=4 \u2192 4-4=0 \u2192 abs=0  \n4+0=4 \u274c expected 3 \u2192 fail\n\nNow try B:  \nx=11: (10*11)//5 = 110//5=22 \u2192 22-3=19 \u2192 +7=26 \u2192 //4=6 \u2192 6-9=-3 \u2192 abs=3  \ny=-6: (5*(-6))//2 = -30//2 = -15 \u2192 -15-6=-21 \u2192 abs=21 \u2192 +3=24 \u2192 //5=4 \u2192 4-4=0 \u2192 abs=0  \n3+0=3 \u2705\n\nx=-9,y=3:  \n(10*(-9))//5 = -90//5 = -18 \u2192 -18-3=-21 \u2192 +7=-14 \u2192 //4 = -4 \u2192 -4-9=-13 \u2192 abs=13  \n(5*3)//2 = 15//2 = 7 \u2192 7-6=1 \u2192 +3=4 \u2192 //5=0 \u2192 0-4=-4 \u2192 abs=4 \u2192 13+4=17 \u274c\n\nStill fails.\n\nTry C: same as B \u2192 same values \u2192 fails\n\nTry D:  \nx=11: (9*11)//5 = 99//5=19 \u2192 19-3=16 \u2192 +7=23 \u2192 //4=5 \u2192 5-9=-4 \u2192 abs=4  \ny=-6: (5*(-6))//2 = -30//2=-15 \u2192 -15-6=-21 \u2192 +3=-18 \u2192 abs=18 \u2192 +3=21 \u2192 //5=4 \u2192 4-4=0 \u2192 abs=0  \n4+0=4 \u274c expected 3\n\nNow check if any expression gives correct output for all.\n\nTry A on x=-9,y=3: we had 7 \u2192 expected 6 \u2192 fail\n\nIs there a candidate that gives 6?\n\nTry to reverse-engineer.\n\nWe need a function f(x,y) that outputs:\n\nx=-4,y=6 \u2192 8  \nx=-9,y=3 \u2192 6  \nx=1,y=8 \u2192 9  \nx=11,y=-6 \u2192 3\n\nTry candidate D on x=-9,y=3:\n\n(9*x)//5 = 9*(-9"
      },
      "penalty": {
        "answer": "A",
        "success": 0,
        "cost": 2049,
        "scratch_budget_hit": true,
        "scratch": "x=-4, y=6:  \nA: abs((abs((9*(-4)//5 - 3) + 7)//4 - 9) + abs((abs((4*6)//2 - 6) + 3)//5 - 4))  \n\u2192 9*(-4) = -36 //5 = -7 \u2192 abs(-7 -3)=10 \u2192 10+7=17 \u2192 17//4=4 \u2192 4-9=-5 \u2192 abs(-5)=5  \n\u2192 4*6=24 //2=12 \u2192 12-6=6 \u2192 abs(6)=6 \u2192 6+3=9 \u2192 9//5=1 \u2192 1-4=-3 \u2192 abs(-3)=3  \n\u2192 5+3=8 \u2705  \n\nx=-9, y=3:  \nA: abs((abs((9*(-9)//5 -3)+7)//4 -9)) + abs((abs((4*3)//2 -6)+3)//5 -4)  \n\u2192 9*(-9) = -81 //5 = -16 \u2192 abs(-16-3)=19 \u2192 19+7=26 \u2192 26//4=6 \u2192 6-9=-3 \u2192 abs=3  \n\u2192 4*3=12 //2=6 \u2192 6-6=0 \u2192 abs=0 \u2192 0+3=3 \u2192 3//5=0 \u2192 0-4=-4 \u2192 abs=4  \n\u2192 3+4=7 \u274c expected 6 \u2192 fail\n\nB: abs((abs((10*(-4)//5 -3)+7)//4 -9)) + abs((abs((5*6)//2 -6)+3)//5 -4)  \n\u2192 10*(-4)=-40 //5=-8 \u2192 abs(-8-3)=11 \u2192 11+7=18 \u2192 18//4=4 \u2192 4-9=-5 \u2192 abs=5  \n\u2192 5*6=30 //2=15 \u2192 15-6=9 \u2192 abs=9 \u2192 9+3=12 \u2192 12//5=2 \u2192 2-4=-2 \u2192 abs=2  \n\u2192 5+2=7 \u274c expected 8 \u2192 fail\n\nC: same as B \u2192 same result \u2192 fail\n\nD: abs((abs((9*(-4)//5 -3)+7)//4 -9)) + abs((abs((5*6)//2 -6)+3)//5 -4)  \n\u2192 9*(-4)=-36 //5=-7 \u2192 abs(-7-3)=10 \u2192 10+7=17 \u2192 17//4=4 \u2192 4-9=-5 \u2192 abs=5  \n\u2192 5*6=30 //2=15 \u2192 15-6=9 \u2192 abs=9 \u2192 9+3=12 \u2192 12//5=2 \u2192 2-4=-2 \u2192 abs=2  \n\u2192 5+2=7 \u274c expected 8 \u2192 fail\n\nA fails at x=-9,y=3 \u2192 not correct.\n\nNo candidate passes all.\n\nWait \u2014 recheck A at x=-9,y=3:\n\nA: abs((abs((9*(-9)//5 -3)+7)//4 -9))  \n9*(-9) = -81 //5 = -16.2 \u2192 floor? Integer division in Python is floor division.  \n-81 // 5 = -17 (since -81 / 5 = -16.2 \u2192 floor is -17)  \n\u2192 -17 -3 = -20 \u2192 abs = 20 \u2192 +7 = 27 \u2192 27//4 = 6 \u2192 6-9 = -3 \u2192 abs = 3  \n\ny part: (4*3)//2 = 12//2 = 6 \u2192 6-6=0 \u2192 abs=0 \u2192 +3=3 \u2192 3//5=0 \u2192 0-4=-4 \u2192 abs=4  \n3+4=7 \u2192 expected 6 \u2192 fail\n\nAll fail?\n\nBut let's check if any matches first example.\n\nx=1,y=8 \u2192 expected 9\n\nTry A:  \nx=1: 9*1=9 //5=1 \u2192 1-3=-2 \u2192 abs=2 \u2192 +7=9 \u2192 9//4=2 \u2192 2-9=-7 \u2192 abs=7  \ny=8: 4*8=32 //2=16 \u2192 16-6=10 \u2192 abs=10 \u2192 +3=13 \u2192 13//5=2 \u2192 2-4=-2 \u2192 abs=2  \n7+2=9 \u2705\n\nx=11,y=-6 \u2192 expected 3\n\nA:  \nx=11: 9*11=99 //5=19 \u2192 19-3=16 \u2192 abs=16 \u2192 +7=23 \u2192 23//4=5 \u2192 5-9=-4 \u2192 abs=4  \ny=-6: 4*(-6)=-24 //2=-12 \u2192 -12-6=-18 \u2192 abs=18 \u2192 +3=21 \u2192 21//5=4 \u2192 4-4=0 \u2192 abs=0  \n4+0=4 \u274c expected 3\n\nSo A fails at x=11,y=-6\n\nB:  \nx=11: 10*11=110 //5=22 \u2192 22-3=19 \u2192 abs=19 \u2192 +7=26 \u2192 26//4=6 \u2192 6-9=-3 \u2192 abs=3  \ny=-6: 5*(-6)=-30 //2=-15 \u2192 -15-6=-21 \u2192 abs=21 \u2192 +3=24 \u2192 24//5=4 \u2192 4-4=0 \u2192 abs=0  \n3+0=3 \u2705\n\nx=1,y=8:  \nB: x part: 10*1=10//5=2 \u2192 2-3=-1 \u2192 abs=1 \u2192 +7=8 \u2192 8//4=2 \u2192 2-9=-7 \u2192 abs=7  \ny part: 5*8=40//2=20 \u2192 20-6=14 \u2192 abs=14 \u2192 +3=17 \u2192 17//5=3 \u2192 3-4=-1 \u2192 abs=1  \n7+1=8 \u274c expected 9 \u2192 fail\n\nC: same as B \u2192 same failure\n\nD:  \nx=1: 9*1=9//5=1 \u2192 1-3=-2 \u2192 abs=2 \u2192 +7=9 \u2192 9//4=2 \u2192 2-9=-7 \u2192 abs=7  \ny=8: 5*8=40//2=20 \u2192 20-6=14 \u2192 abs=14 \u2192 +3=17 \u2192 17//5=3 \u2192 3-4=-1 \u2192 abs=1  \n7+1=8 \u274c expected 9 \u2192 fail\n\nOnly B passed x=11,y=-6, but failed x=1,y=8.\n\nIs there a candidate that works on first three?\n\nx=-4,y=6 \u2192 expected 8\n\nTry B:  \nx=-4: 10*(-4)=-40//5=-8 \u2192 -8-3=-11 \u2192 abs=11 \u2192 +7=18 \u2192 18//4=4 \u2192 4-9=-5 \u2192 abs=5  \ny=6: 5*6=30//2=15 \u2192 15-6=9 \u2192 abs=9 \u2192 +3=12 \u2192 12//5=2 \u2192 2-4=-2 \u2192 abs=2  \n5+2=7 \u274c expected 8\n\nNo.\n\nWait \u2014 maybe I made a mistake in the y part of D.\n\nD: y part: (abs((5*y)//2 -6) +3)//5 -4\n\nAt y=6: (5*6)=30//2=15 \u2192 15-6=9 \u2192 abs=9 \u2192 +3=12 \u2192 12//5=2 \u2192 2-4=-2 \u2192 abs=2 \u2192 so part is 2\n\nx part: 9*(-4)=-36//5 = -7.2 \u2192 floor to -8? -36//5 = -8?  \nIn Python: -36 // 5 = -8 (since 5*(-8) = -40, which is less than -36) \u2192 correct  \n-8 -3 = -11 \u2192 abs=11 \u2192 +7=18 \u2192 18//4=4 \u2192 4-9=-5 \u2192 abs=5  \n5+2=7 \u2192 not 8\n\nAll fail?\n\nBut must be one.\n\nTry candidate C and D with x=1,y=8.\n\nC: x=1: (10*1//5)=2 \u2192 2-3=-1 \u2192 abs=1 \u2192 +7=8 \u2192 8//4=2 \u2192 2-9=-7 \u2192 abs=7  \ny=8: (5*8//2)=40//2"
      }
    },
    {
      "outcome_cell": "01",
      "task_id": "v3-t3-dev-00004",
      "sample": 0,
      "seed": 36261405,
      "family": "nested_absolute",
      "candidates": [
        "abs(abs(abs(abs(x - 2) - 8) - 7) - 6) + abs(abs(abs(abs(y - 7) - 6) - 4) - 3)",
        "abs(abs(abs(abs(x + 2) - 8) - 7) - 6) + abs(abs(abs(abs(y - 7) - 6) - 4) - 3)",
        "abs(abs(abs(abs(x - 2) - 8) - 7) - 6) + abs(abs(abs(abs(y + 7) - 6) - 4) - 3)",
        "abs(abs(abs(abs(x + 2) - 8) - 7) - 6) + abs(abs(abs(abs(y + 7) - 6) - 4) - 3)"
      ],
      "target": "abs(abs(abs(abs(x + 2) - 8) - 7) - 6) + abs(abs(abs(abs(y - 7) - 6) - 4) - 3)",
      "visible_inputs": [
        [
          7,
          -3
        ],
        [
          -15,
          11
        ],
        [
          15,
          6
        ],
        [
          14,
          7
        ]
      ],
      "visible_examples": [
        {
          "x": 7,
          "y": -3,
          "output": 3
        },
        {
          "x": -15,
          "y": 11,
          "output": 5
        },
        {
          "x": 15,
          "y": 6,
          "output": 6
        },
        {
          "x": 14,
          "y": 7,
          "output": 6
        }
      ],
      "control": {
        "answer": "A",
        "success": 0,
        "cost": 2049,
        "scratch_budget_hit": true,
        "scratch": "x=7, y=-3 \u2192 3  \nx=-15, y=11 \u2192 5  \nx=15, y=6 \u2192 6  \nx=14, y=7 \u2192 6  \n\nTest A:  \nx=7, y=-3:  \nabs(abs(abs(abs(7-2)-8)-7)-6) = abs(abs(abs(5-8)-7)-6) = abs(abs(3-7)-6) = abs(4-6) = 2  \nabs(abs(abs(abs(-3-7)-6)-4)-3) = abs(abs(abs(-10-6)-6)-4)-3) = abs(abs(-16-6)-4)-3) = abs(-22-4)-3 = abs(-26)-3 = 26-3 = 23  \nTotal: 2 + 23 = 25 \u2192 not 3 \u2192 fail\n\nTest B:  \nx=7, y=-3:  \nabs(abs(abs(abs(7+2)-8)-7)-6) = abs(abs(abs(9-8)-7)-6) = abs(abs(1-7)-6) = abs(6-6) = 0  \nabs(abs(abs(abs(-3-7)-6)-4)-3) = abs(abs(abs(-10-6)-6)-4)-3) = abs(abs(-16-6)-4)-3) = abs(-22-4)-3 = 26-3 = 23  \nTotal: 0 + 23 = 23 \u2192 not 3 \u2192 fail\n\nTest C:  \nx=7, y=-3:  \nabs(abs(abs(abs(7-2)-8)-7)-6) = abs(abs(abs(5-8)-7)-6) = abs(abs(3-7)-6) = abs(4-6) = 2  \nabs(abs(abs(abs(-3+7)-6)-4)-3) = abs(abs(abs(4-6)-4)-3) = abs(abs(-2-4)-3) = abs(-6-3) = 9  \nTotal: 2 + 9 = 11 \u2192 not 3 \u2192 fail\n\nTest D:  \nx=7, y=-3:  \nabs(abs(abs(abs(7+2)-8)-7)-6) = abs(abs(abs(9-8)-7)-6) = abs(abs(1-7)-6) = abs(6-6) = 0  \nabs(abs(abs(abs(-3+7)-6)-4)-3) = abs(abs(abs(4-6)-4)-3) = abs(abs(-2-4)-3) = abs(-6-3) = 9  \nTotal: 0 + 9 = 9 \u2192 not 3 \u2192 fail\n\nAll candidates fail? But problem says \"only one\" satisfies every pair.\n\nWait \u2014 recheck the first example: x=7, y=-3 \u2192 3\n\nTry to reverse engineer.\n\nTry candidate A again, more carefully:\n\nx=7, y=-3  \nx-2 = 5  \nabs(5-8)=3  \nabs(3-7)=4  \nabs(4-6)=2  \n\ny-7 = -10  \nabs(-10-6)=16  \nabs(16-4)=12  \nabs(12-3)=9 \u2192 too big\n\nWait \u2014 maybe I misread the structure.\n\nAll expressions are of form:  \nabs(abs(abs(abs(x \u00b1 a) - b) - c) - d) + same for y with similar constants\n\nTry to find expression that gives 3 for (7,-3)\n\nTry candidate D:  \nx=7+2=9 \u2192 abs(9-8)=1 \u2192 abs(1-7)=6 \u2192 abs(6-6)=0 \u2192 abs(0-6)=6? Wait no:\n\nWait expression D:  \nabs(abs(abs(abs(x+2)-8)-7)-6) \u2192  \nx=7: x+2=9 \u2192 abs(9-8)=1 \u2192 abs(1-7)=6 \u2192 abs(6-6)=0\n\ny=-3: y+7=4 \u2192 abs(4-6)=2 \u2192 abs(2-4)=2 \u2192 abs(2-3)=1\n\nWait! I made a mistake earlier \u2014 the y part is:  \nabs(abs(abs(abs(y+7)-6)-4)-3)\n\ny=-3 \u2192 y+7=4  \nabs(4-6)=2  \nabs(2-4)=2  \nabs(2-3)=1 \u2192 not 1, wait: abs(2-4)=2, then abs(2-3)=1 \u2192 total = 0 + 1 = 1 \u2192 not 3\n\nStill not 3.\n\nWait \u2014 maybe I misread the expression.\n\nLet me write it again:\n\nA: abs(abs(abs(abs(x-2)-8)-7)-6) + abs(abs(abs(abs(y-7)-6)-4)-3)\n\nB: abs(abs(abs(abs(x+2)-8)-7)-6) + abs(abs(abs(abs(y-7)-6)-4)-3)\n\nC: abs(abs(abs(abs(x-2)-8)-7)-6) + abs(abs(abs(abs(y+7)-6)-4)-3)\n\nD: abs(abs(abs(abs(x+2)-8)-7)-6) + abs(abs(abs(abs(y+7)-6)-4)-3)\n\nTry A with x=7, y=-3:\n\nx-2 = 5  \nabs(5-8)=3  \nabs(3-7)=4  \nabs(4-6)=2\n\ny-7 = -10  \nabs(-10-6)=16  \nabs(16-4)=12  \nabs(12-3)=9  \nSum: 2+9=11 \u2192 no\n\nTry B:  \nx=7: x+2=9 \u2192 abs(9-8)=1 \u2192 abs(1-7)=6 \u2192 abs(6-6)=0  \ny=-3: y-7=-10 \u2192 abs(-10-6)=16 \u2192 abs(16-4)=12 \u2192 abs(12-3)=9 \u2192 sum=0+9=9 \u2192 no\n\nC:  \nx=7: x-2=5 \u2192 abs(5-8)=3 \u2192 abs(3-7)=4 \u2192 abs(4-6)=2  \ny=-3: y+7=4 \u2192 abs(4-6)=2 \u2192 abs(2-4)=2 \u2192 abs(2-3)=1 \u2192 sum=2+1=3 \u2192 YES!\n\nNow check second pair: x=-15, y=11 \u2192 should be 5\n\nC:  \nx=-15: x-2 = -17  \nabs(-17-8)=25  \nabs(25-7)=18  \nabs(18-6)=12  \n\u2192 12\n\ny=11: y+7=18  \nabs(18-6)=12  \nabs(12-4)=8  \nabs(8-3)=5  \n\u2192 5  \nSum: 12+5=17 \u2192 not 5 \u2192 fail\n\nWait \u2014 no.\n\nWait \u2014 the expression is:  \nabs(abs(abs(abs(x-2)-8)-7)-6) \u2192 so nested abs\n\nx=-15:  \nx-2 = -17  \nabs(-17) = 17  \n17 - 8 = 9 \u2192 abs(9)=9  \n9 - 7 = 2 \u2192 abs(2)=2  \n2 - 6 = -4 \u2192 abs(-4)=4  \n\u2192 4\n\ny=11: y+7=18  \nabs(18-6)=12  \nabs(12-4)=8  \nabs(8-3)=5  \n\u2192 5  \nSum: 4+5=9 \u2192 not 5 \u2192 fail\n\nNo.\n\nTry D: x=7, y=-3\n\nx=7: x+2=9 \u2192 abs(9-8)=1 \u2192 abs(1-7)=6 \u2192 abs(6-6)=0  \ny=-3: y+7=4 \u2192 abs(4-6)=2 \u2192 abs(2-4)=2 \u2192 abs(2-3)=1 \u2192 sum=0+1=1 \u2192 not 3\n\nNo.\n\nTry B with x=7, y=-3:  \nx=7: x+2=9 \u2192 abs(9-8)=1 \u2192 abs(1-7)=6 \u2192 abs(6-6)=0  \ny=-3: y-7=-10 \u2192 abs(-10-6)=16 \u2192 abs(16-4)=12 \u2192 abs(12-3)=9 \u2192 sum=0+9=9 \u2192 no\n\nA: x=7,y=-3 \u2192 x part: 2, y part: y-7=-10 \u2192 abs(-10-6)=16 \u2192 abs(16-4)=12 \u2192 abs(12-3)=9 \u2192 sum=11 \u2192 no\n\nWait \u2014 perhaps I made a mistake in the y part.\n\nLet\u2019s redefine the structure.\n\nAll expressions have the form:  \nouter: abs( ... )  \nfour levels: | | | | val - c | - d | - e | - f |\n\nSo for x: | | | | x \u00b1 a | - b | - c | - d | - e |\n\nFor each, compute step by"
      },
      "penalty": {
        "answer": "B",
        "success": 1,
        "cost": 2049,
        "scratch_budget_hit": true,
        "scratch": "x=7, y=-3 \u2192 3  \nx=-15, y=11 \u2192 5  \nx=15, y=6 \u2192 6  \nx=14, y=7 \u2192 6  \n\nTest A:  \nx=7, y=-3:  \nabs(abs(abs(abs(7-2)-8)-7)-6) = abs(abs(abs(5-8)-7)-6) = abs(abs(3-7)-6) = abs(4-6) = 2  \nabs(abs(abs(abs(-3-7)-6)-4)-3) = abs(abs(abs(-10-6)-6)-4)-3) = abs(abs(16-6)-4)-3) = abs(10-4)-3 = 6-3 = 3  \nTotal: 2+3 = 5 \u2192 should be 3 \u2192 mismatch  \n\nFail A.\n\nTest B:  \nx=7, y=-3:  \nabs(abs(abs(abs(7+2)-8)-7)-6) = abs(abs(abs(9-8)-7)-6) = abs(abs(1-7)-6) = abs(6-6) = 0  \nabs(abs(abs(abs(-3-7)-6)-4)-3) = abs(abs(abs(-10-6)-6)-4)-3) = abs(abs(16-6)-4)-3) = abs(10-4)-3 = 6-3 = 3  \nTotal: 0+3 = 3 \u2192 good  \n\nx=-15, y=11:  \nabs(abs(abs(abs(-15+2)-8)-7)-6) = abs(abs(abs(-13-8)-7)-6) = abs(abs(-21-7)-6) = abs(-28-6) = 34  \nWait \u2014 too large. Let's step carefully:  \nabs(-15+2) = 13  \nabs(13-8)=5  \nabs(5-7)=2  \nabs(2-6)=4  \n\u2192 first part: 4  \n\ny=11:  \nabs(abs(abs(abs(11-7)-6)-4)-3) = abs(abs(abs(4-6)-4)-3) = abs(abs(2-4)-3) = abs(2-3)=1  \nTotal: 4+1 = 5 \u2192 good  \n\nx=15, y=6:  \nx part: abs(abs(abs(abs(15+2)-8)-7)-6) = abs(abs(abs(17-8)-7)-6) = abs(abs(9-7)-6)=abs(2-6)=4  \ny part: abs(abs(abs(abs(6-7)-6)-4)-3) = abs(abs(abs(-1-6)-4)-3)=abs(abs(7-4)-3)=abs(3-3)=0  \nTotal: 4+0=4 \u2192 should be 6 \u2192 mismatch  \n\nFail B.\n\nTest C:  \nx=7, y=-3:  \nx part: abs(abs(abs(abs(7-2)-8)-7)-6) = abs(abs(abs(5-8)-7)-6)=abs(abs(3-7)-6)=abs(4-6)=2  \ny part: abs(abs(abs(abs(-3+7)-6)-4)-3)=abs(abs(abs(4-6)-4)-3)=abs(abs(2-4)-3)=abs(2-3)=1  \nTotal: 2+1=3 \u2192 good  \n\nx=-15, y=11:  \nx part: same as A: abs(abs(abs(abs(-15-2)-8)-7)-6) = abs(abs(abs(-17-8)-7)-6)=abs(abs(-25-7)-6)=abs(-32-6)=38 \u2192 too big  \nWait: step by step:  \nx=-15: abs(-15-2)=17  \nabs(17-8)=9  \nabs(9-7)=2  \nabs(2-6)=4 \u2192 x part = 4  \n\ny=11: abs(abs(abs(abs(11+7)-6)-4)-3)=abs(abs(abs(18-6)-4)-3)=abs(abs(12-4)-3)=abs(8-3)=5  \nTotal: 4+5=9 \u2192 should be 5 \u2192 mismatch  \n\nFail C.\n\nTest D:  \nx=7, y=-3:  \nx: abs(abs(abs(abs(7+2)-8)-7)-6)=abs(abs(abs(9-8)-7)-6)=abs(abs(1-7)-6)=abs(6-6)=0  \ny: abs(abs(abs(abs(-3+7)-6)-4)-3)=abs(abs(abs(4-6)-4)-3)=abs(abs(2-4)-3)=abs(2-3)=1  \nTotal: 0+1=1 \u2192 should be 3 \u2192 mismatch  \n\nFail D.\n\nAll fail? But must have one correct.\n\nWait \u2014 recheck B.\n\nx=15, y=6 \u2192 should be 6  \nB:  \nx=15: abs(abs(abs(abs(15+2)-8)-7)-6)  \n17-8=9 \u2192 9-7=2 \u2192 2-6 = -4 \u2192 abs=4  \ny=6: abs(abs(abs(abs(6-7)-6)-4)-3) = abs(abs(abs(-1-6)-4)-3)=abs(abs(7-4)-3)=abs(3-3)=0  \n4+0=4 \u2260 6 \u2192 fail\n\nWait \u2014 perhaps I miscalculated the pattern.\n\nLook at the outputs:  \nx=7,y=-3 \u2192 3  \nx=-15,y=11 \u2192 5  \nx=15,y=6 \u2192 6  \nx=14,y=7 \u2192 6\n\nTry to reverse-engineer.\n\nTry A again with x=7,y=-3:\n\nx: abs(abs(abs(abs(7-2)-8)-7)-6)  \n5-8=-3 \u2192 abs=3  \n3-7=-4 \u2192 abs=4  \n4-7=-3 \u2192 abs=3  \n3-6=-3 \u2192 abs=3 \u2192 x part = 3\n\ny: abs(abs(abs(abs(-3-7)-6)-4)-3)  \n-10-6=-16 \u2192 abs=16  \n16-6=10  \n10-4=6  \n6-3=3 \u2192 y part =3  \nTotal: 3+3=6 \u2192 should be 3 \u2192 fail\n\nWait \u2014 earlier I messed up.\n\nCorrect step-by-step for A:\n\nx=7:  \nx-2=5  \nabs(5)=5  \n5-8=-3 \u2192 abs=3  \n3-7=-4 \u2192 abs=4  \n4-6=-2 \u2192 abs=2 \u2192 x part = 2\n\ny=-3:  \ny-7=-10 \u2192 abs=10  \n10-6=4  \n4-4=0  \n0-3=-3 \u2192 abs=3 \u2192 y part=3  \nTotal: 2+3=5 \u2192 should be 3 \u2192 fail\n\nTry B: x=7,y=-3  \nx+2=9  \n9-8=1  \n1-7=-6 \u2192 abs=6  \n6-6=0 \u2192 x part=0  \ny: -3-7=-10 \u219210\u219210-6=4\u21924-4=0\u21920-3=-3\u21923 \u2192 y part=3  \nTotal: 0+3=3 \u2192 good\n\nx=-15,y=11  \nx+2=-13 \u2192 abs=13  \n13-8=5  \n5-7=-2 \u2192 abs=2  \n2-6=-4 \u2192 abs=4 \u2192 x part=4  \ny=11: 11-7=4 \u21924-6=-2 \u2192 abs=2 \u21922-4=-2 \u2192 abs=2 \u21922-3=-1 \u2192 abs=1 \u2192 y part=1  \nTotal: 4+1=5 \u2192 good\n\nx=15,y=6  \nx+2=17 \u219217-8=9 \u21929-7=2 \u21922-6=-4 \u2192 abs=4  \ny=6: 6-7=-1 \u21921-6=-5 \u21925-4=1 \u21921-3=-2 \u2192 abs=2  \nTotal: 4+2=6 \u2192 good\n\nx=14,y=7  \nx+2=16 \u219216-8=8 \u21928-7=1 \u21921-6=-5 \u2192 abs=5  \ny=7: 7-7=0 \u21920-6=-6 \u21926-4=2 \u21922-3=-1 \u2192 abs=1  \nTotal: 5+1=6 \u2192 good\n\nAll match!\n\nSo B works.\n\nEarlier I made a mistake in y part of B for x=15,y=6.\n\nCorrect y part:  \nabs(abs(abs(abs(y-7)-6)-4)-3)  \ny=6: 6-7=-1 \u2192 abs=1  \n1-6=-5 \u2192 abs=5  \n5-4=1 \u21921-3=-2 \u2192 abs=2 \u2192 yes, 2\n\nx part: 15+2=17 \u219217-8=9 \u21929-7=2"
      }
    },
    {
      "outcome_cell": "10",
      "task_id": "v3-t3-dev-00001",
      "sample": 0,
      "seed": 36261105,
      "family": "bounded_affine",
      "candidates": [
        "min(32, max(0, min(35, max(0, 5 * x - 4) + 2) - 5) + 5) + min(72, max(0, min(49, max(0, 7 * y - 9) + 8) - 3) + 7)",
        "min(32, max(0, min(35, max(0, 5 * x - 4) + 2) - 5) + 5) + min(72, max(0, min(49, max(0, 8 * y - 9) + 8) - 3) + 7)",
        "min(32, max(0, min(35, max(0, 6 * x - 4) + 2) - 5) + 5) + min(72, max(0, min(49, max(0, 7 * y - 9) + 8) - 3) + 7)",
        "min(32, max(0, min(35, max(0, 6 * x - 4) + 2) - 5) + 5) + min(72, max(0, min(49, max(0, 8 * y - 9) + 8) - 3) + 7)"
      ],
      "target": "min(32, max(0, min(35, max(0, 5 * x - 4) + 2) - 5) + 5) + min(72, max(0, min(49, max(0, 7 * y - 9) + 8) - 3) + 7)",
      "visible_inputs": [
        [
          13,
          4
        ],
        [
          7,
          3
        ],
        [
          2,
          9
        ],
        [
          2,
          -8
        ]
      ],
      "visible_examples": [
        {
          "x": 13,
          "y": 4,
          "output": 63
        },
        {
          "x": 7,
          "y": 3,
          "output": 56
        },
        {
          "x": 2,
          "y": 9,
          "output": 61
        },
        {
          "x": 2,
          "y": -8,
          "output": 20
        }
      ],
      "control": {
        "answer": "A",
        "success": 1,
        "cost": 1704,
        "scratch_budget_hit": false,
        "scratch": "x=13, y=4 \u2192 63  \nx=7, y=3 \u2192 56  \nx=2, y=9 \u2192 61  \nx=2, y=-8 \u2192 20  \n\nTest candidate A:  \nx=13, y=4  \nFirst part: min(32, max(0, min(35, max(0, 5*13 - 4) + 2) - 5) + 5)  \n5*13 - 4 = 65 - 4 = 61 \u2192 max(0,61) = 61  \n61 + 2 = 63 \u2192 min(35,63) = 35  \n35 - 5 = 30 \u2192 max(0,30) = 30  \n30 + 5 = 35 \u2192 min(32,35) = 32  \n\nSecond part: min(72, max(0, min(49, max(0, 7*4 - 9) + 8) - 3) + 7)  \n7*4 - 9 = 28 - 9 = 19 \u2192 max(0,19) = 19  \n19 + 8 = 27 \u2192 min(49,27) = 27  \n27 - 3 = 24 \u2192 max(0,24) = 24  \n24 + 7 = 31 \u2192 min(72,31) = 31  \n\nTotal: 32 + 31 = 63 \u2192 matches  \n\nx=7, y=3  \nFirst: 5*7 - 4 = 35 - 4 = 31 \u2192 max(0,31)=31 \u2192 +2 = 33 \u2192 min(35,33)=33 \u2192 -5 = 28 \u2192 +5 = 33 \u2192 min(32,33)=32  \nSecond: 7*3 - 9 = 21 - 9 = 12 \u2192 +8 = 20 \u2192 min(49,20)=20 \u2192 -3 = 17 \u2192 +7 = 24 \u2192 min(72,24)=24  \nTotal: 32 + 24 = 56 \u2192 matches  \n\nx=2, y=9  \nFirst: 5*2 - 4 = 10 - 4 = 6 \u2192 +2 = 8 \u2192 min(35,8)=8 \u2192 -5 = 3 \u2192 +5 = 8 \u2192 min(32,8)=8  \nSecond: 7*9 - 9 = 63 - 9 = 54 \u2192 +8 = 62 \u2192 min(49,62)=49 \u2192 -3 = 46 \u2192 +7 = 53 \u2192 min(72,53)=53  \nTotal: 8 + 53 = 61 \u2192 matches  \n\nx=2, y=-8  \nFirst: 5*2 - 4 = 6 \u2192 +2 = 8 \u2192 min(35,8)=8 \u2192 -5 = 3 \u2192 +5 = 8 \u2192 min(32,8)=8  \nSecond: 7*(-8) - 9 = -56 - 9 = -65 \u2192 max(0,-65)=0 \u2192 +8 = 8 \u2192 min(49,8)=8 \u2192 -3 = 5 \u2192 +7 = 12 \u2192 min(72,12)=12  \nTotal: 8 + 12 = 20 \u2192 matches  \n\nA passes all.\n\nNow check if any other candidate fails earlier \u2014 stop at first mismatch.\n\nTest candidate B:  \nx=13, y=4  \nFirst part: min(32, max(0, min(35, max(0, 5*13 - 4) + 2) - 5) + 5)  \nSame as A: 5*13-4=61 \u2192 +2=63 \u2192 min(35,63)=35 \u2192 -5=30 \u2192 +5=35 \u2192 min(32,35)=32  \nSecond part: min(72, max(0, min(49, max(0, 8*4 - 9) + 8) - 3) + 7)  \n8*4 - 9 = 32 - 9 = 23 \u2192 +8 = 31 \u2192 min(49,31)=31 \u2192 -3 = 28 \u2192 +7 = 35 \u2192 min(72,35)=35  \nTotal: 32 + 35 = 67 \u2260 63 \u2192 fails  \n\nB fails at first example.\n\nC:  \nx=13, y=4  \nFirst: 6*13 - 4 = 78 - 4 = 74 \u2192 max(0,74)=74 \u2192 +2 = 76 \u2192 min(35,76)=35 \u2192 -5 = 30 \u2192 +5 = 35 \u2192 min(32,35)=32  \nSecond: 7*4 - 9 = 19 \u2192 +8 = 27 \u2192 min(49,27)=27 \u2192 -3 = 24 \u2192 +7 = 31 \u2192 min(72,31)=31  \nTotal: 32 + 31 = 63 \u2192 matches  \n\nx=7, y=3  \nFirst: 6*7 - 4 = 42 - 4 = 38 \u2192 +2 = 40 \u2192 min(35,40)=35 \u2192 -5 = 30 \u2192 +5 = 35 \u2192 min(32,35)=32  \nSecond: 7*3 - 9 = 12 \u2192 +8 = 20 \u2192 min(49,20)=20 \u2192 -3 = 17 \u2192 +7 = 24 \u2192 min(72,24)=24  \nTotal: 32 + 24 = 56 \u2192 matches  \n\nx=2, y=9  \nFirst: 6*2 - 4 = 12 - 4 = 8 \u2192 +2 = 10 \u2192 min(35,10)=10 \u2192 -5 = 5 \u2192 +5 = 10 \u2192 min(32,10)=10  \nSecond: 7*9 - 9 = 54 \u2192 +8 = 62 \u2192 min(49,62)=49 \u2192 -3 = 46 \u2192 +7 = 53 \u2192 min(72,53)=53  \nTotal: 10 + 53 = 63 \u2260 61 \u2192 fails  \n\nC fails at x=2,y=9  \n\nD:  \nx=13, y=4  \nFirst: 6*13 - 4 = 78 - 4 = 74 \u2192 +2 = 76 \u2192 min(35,76)=35 \u2192 -5 = 30 \u2192 +5 = 35 \u2192 min(32,35)=32  \nSecond: 8*4 - 9 = 32 - 9 = 23 \u2192 +8 = 31 \u2192 min(49,31)=31 \u2192 -3 = 28 \u2192 +7 = 35 \u2192 min(72,35)=35  \nTotal: 32 + 35 = 67 \u2260 63 \u2192 fails  \n\nOnly A passes all examples.\n\nThe correct repair is A."
      },
      "penalty": {
        "answer": "C",
        "success": 0,
        "cost": 2049,
        "scratch_budget_hit": true,
        "scratch": "x=13, y=4 \u2192 63  \nx=7, y=3 \u2192 56  \nx=2, y=9 \u2192 61  \nx=2, y=-8 \u2192 20  \n\nTest A:  \nx=13, y=4  \n5*x - 4 = 65 - 4 = 61 \u2192 max(0,61)=61 \u2192 +2 = 63 \u2192 min(35,63)=35 \u2192 -5 = 30 \u2192 +5 = 35 \u2192 min(32,35)=32  \n8*y - 9 = 32 - 9 = 23 \u2192 max(0,23)=23 \u2192 +8 = 31 \u2192 min(49,31)=31 \u2192 -3 = 28 \u2192 +7 = 35 \u2192 min(72,35)=35  \nTotal: 32 + 35 = 67 \u2192 not 63 \u2192 fail  \n\nTest B:  \nx=13, y=4  \n5*x - 4 = 61 \u2192 max(0,61)=61 \u2192 +2 = 63 \u2192 min(35,63)=35 \u2192 -5 = 30 \u2192 +5 = 35 \u2192 min(32,35)=32  \n8*y - 9 = 32 - 9 = 23 \u2192 max(0,23)=23 \u2192 +8 = 31 \u2192 min(49,31)=31 \u2192 -3 = 28 \u2192 +7 = 35 \u2192 min(72,35)=35  \n32 + 35 = 67 \u2192 not 63 \u2192 fail  \n\nTest C:  \nx=13, y=4  \n6*x - 4 = 78 - 4 = 74 \u2192 max(0,74)=74 \u2192 +2 = 76 \u2192 min(35,76)=35 \u2192 -5 = 30 \u2192 +5 = 35 \u2192 min(32,35)=32  \n7*y - 9 = 28 - 9 = 19 \u2192 max(0,19)=19 \u2192 +8 = 27 \u2192 min(49,27)=27 \u2192 -3 = 24 \u2192 +7 = 31 \u2192 min(72,31)=31  \n32 + 31 = 63 \u2192 good  \nx=7, y=3  \n6*x - 4 = 42 - 4 = 38 \u2192 max(0,38)=38 \u2192 +2 = 40 \u2192 min(35,40)=35 \u2192 -5 = 30 \u2192 +5 = 35 \u2192 min(32,35)=32  \n7*y - 9 = 21 - 9 = 12 \u2192 max(0,12)=12 \u2192 +8 = 20 \u2192 min(49,20)=20 \u2192 -3 = 17 \u2192 +7 = 24 \u2192 min(72,24)=24  \n32 + 24 = 56 \u2192 good  \nx=2, y=9  \n6*x - 4 = 12 - 4 = 8 \u2192 max(0,8)=8 \u2192 +2 = 10 \u2192 min(35,10)=10 \u2192 -5 = 5 \u2192 +5 = 10 \u2192 min(32,10)=10  \n7*y - 9 = 63 - 9 = 54 \u2192 max(0,54)=54 \u2192 +8 = 62 \u2192 min(49,62)=49 \u2192 -3 = 46 \u2192 +7 = 53 \u2192 min(72,53)=53  \n10 + 53 = 63 \u2192 but expected 61 \u2192 fail  \n\nTest D:  \nx=13, y=4  \n6*x - 4 = 74 \u2192 max=74 \u2192 +2=76 \u2192 min(35,76)=35 \u2192 -5=30 \u2192 +5=35 \u2192 min(32,35)=32  \n8*y - 9 = 32 - 9 = 23 \u2192 max=23 \u2192 +8=31 \u2192 min(49,31)=31 \u2192 -3=28 \u2192 +7=35 \u2192 min(72,35)=35  \n32 + 35 = 67 \u2192 not 63 \u2192 fail  \n\nAll candidates fail at least one example. Wait \u2014 did I miscalculate?\n\nRecheck C: x=2, y=9 \u2192 expected 61  \n6*x - 4 = 12 - 4 = 8 \u2192 max(0,8)=8 \u2192 +2 = 10 \u2192 min(35,10)=10 \u2192 -5 = 5 \u2192 +5 = 10 \u2192 min(32,10)=10  \n7*y - 9 = 63 - 9 = 54 \u2192 max(0,54)=54 \u2192 +8 = 62 \u2192 min(49,62)=49 \u2192 -3 = 46 \u2192 +7 = 53 \u2192 min(72,53)=53  \n10 + 53 = 63 \u2192 not 61 \u2192 fail  \n\nIs there a candidate that gives 61?\n\nWait \u2014 maybe I misread the expressions.\n\nLet me go back to candidate C and D \u2014 perhaps the y part is different.\n\nWait \u2014 in C and D, y uses 7*y or 8*y.\n\nTry candidate C again with x=2, y=9 \u2192 output 63 \u2192 expected 61 \u2192 too high.\n\nTry x=2, y=-8 \u2192 for all?\n\nTry candidate A with x=2, y=-8  \n5*x - 4 = 10 - 4 = 6 \u2192 max(0,6)=6 \u2192 +2 = 8 \u2192 min(35,8)=8 \u2192 -5 = 3 \u2192 +5 = 8 \u2192 min(32,8)=8  \n8*y - 9 = -64 -9 = -73 \u2192 max(0,-73)=0 \u2192 +8 = 8 \u2192 min(49,8)=8 \u2192 -3 = 5 \u2192 +7 = 12 \u2192 min(72,12)=12  \n8 + 12 = 20 \u2192 matches  \nx=13, y=4 \u2192 earlier gave 67 \u2192 not 63 \u2192 fail  \n\nCandidate B: x=2, y=-8  \n5*x-4=6 \u2192 max(0,6)=6 \u2192 +2=8 \u2192 min(35,8)=8 \u2192 -5=3 \u2192 +5=8 \u2192 min(32,8)=8  \n8*y-9 = -64-9=-73 \u2192 max(0,-73)=0 \u2192 +8=8 \u2192 min(49,8)=8 \u2192 -3=5 \u2192 +7=12 \u2192 min(72,12)=12  \n8+12=20 \u2192 good  \nx=13, y=4:  \n5*x-4=61 \u2192 max=61 \u2192 +2=63 \u2192 min(35,63)=35 \u2192 -5=30 \u2192 +5=35 \u2192 min(32,35)=32  \n8*y-9=32-9=23 \u2192 max=23 \u2192 +8=31 \u2192 min(49,31)=31 \u2192 -3=28 \u2192 +7=35 \u2192 min(72,35)=35  \n32+35=67 \u226063 \u2192 fail  \n\nCandidate D: x=2, y=-8  \n6*x-4=12-4=8 \u2192 max=8 \u2192 +2=10 \u2192 min(35,10)=10 \u2192 -5=5 \u2192 +5=10 \u2192 min(32,10)=10  \n8*y-9=-64-9=-73 \u2192 max=0 \u2192 +8=8 \u2192 min(49,8)=8 \u2192 -3=5 \u2192 +7=12 \u2192 min(72,12)=12  \n10+12=22 \u2192 not 20 \u2192 fail  \n\nSo only candidate C and D have 7y in y part, others have 8y.\n\nWait \u2014 maybe I miscalculated x=2, y=9 for C.\n\nC: x=2, y=9  \n6x-4 = 12-4=8 \u2192 +2=10 \u2192 min(35,10)=10 \u2192 -5=5 \u2192 +5=10 \u2192 min(32,10)=10  \n7y-9 = 63-9=54 \u2192 +8=62 \u2192 min(49,62)=49 \u2192 -3=46 \u2192 +7=53 \u2192 min(72,53)=53  \n10+53=63 \u2192 expected 61 \u2192 off by 2  \n\nWhat if the y part"
      }
    },
    {
      "outcome_cell": "11",
      "task_id": "v3-t3-dev-00000",
      "sample": 0,
      "seed": 36261005,
      "family": "nested_absolute",
      "candidates": [
        "abs(abs(abs(abs(x - 4) - 9) - 2) - 4) + abs(abs(abs(abs(y - 5) - 7) - 8) - 2)",
        "abs(abs(abs(abs(x + 4) - 9) - 2) - 4) + abs(abs(abs(abs(y - 5) - 7) - 8) - 2)",
        "abs(abs(abs(abs(x - 4) - 9) - 2) - 4) + abs(abs(abs(abs(y + 5) - 7) - 8) - 2)",
        "abs(abs(abs(abs(x + 4) - 9) - 2) - 4) + abs(abs(abs(abs(y + 5) - 7) - 8) - 2)"
      ],
      "target": "abs(abs(abs(abs(x - 4) - 9) - 2) - 4) + abs(abs(abs(abs(y - 5) - 7) - 8) - 2)",
      "visible_inputs": [
        [
          -15,
          -16
        ],
        [
          13,
          -3
        ],
        [
          0,
          12
        ],
        [
          -12,
          -1
        ]
      ],
      "visible_examples": [
        {
          "x": -15,
          "y": -16,
          "output": 8
        },
        {
          "x": 13,
          "y": -3,
          "output": 7
        },
        {
          "x": 0,
          "y": 12,
          "output": 7
        },
        {
          "x": -12,
          "y": -1,
          "output": 6
        }
      ],
      "control": {
        "answer": "A",
        "success": 1,
        "cost": 1094,
        "scratch_budget_hit": false,
        "scratch": "x=-15, y=-16 \u2192 8  \nx=13, y=-3 \u2192 7  \nx=0, y=12 \u2192 7  \nx=-12, y=-1 \u2192 6\n\nTest A:  \nx=-15, y=-16  \nabs(-15 - 4) = 19  \nabs(19 - 9) = 10  \nabs(10 - 2) = 8  \nabs(8 - 4) = 4  \n\u2192 4  \n\ny: abs(-16 - 5) = 21  \nabs(21 - 7) = 14  \nabs(14 - 8) = 6  \nabs(6 - 2) = 4  \n\u2192 4  \nTotal: 4 + 4 = 8 \u2705\n\nx=13, y=-3  \nx: abs(13-4)=9 \u2192 9-9=0 \u2192 0-2=-2 \u2192 abs=2 \u2192 2-4=-2 \u2192 abs=2  \ny: abs(-3-5)=8 \u2192 8-7=1 \u2192 1-8=-7 \u2192 abs=7 \u2192 7-2=5 \u2192 abs=5  \nTotal: 2 + 5 = 7 \u2705\n\nx=0, y=12  \nx: abs(0-4)=4 \u2192 4-9=-5 \u2192 abs=5 \u2192 5-2=3 \u2192 3-4=-1 \u2192 abs=1  \ny: abs(12-5)=7 \u2192 7-7=0 \u2192 0-8=-8 \u2192 abs=8 \u2192 8-2=6  \nTotal: 1 + 6 = 7 \u2705\n\nx=-12, y=-1  \nx: abs(-12-4)=16 \u2192 16-9=7 \u2192 7-2=5 \u2192 5-4=1 \u2192 abs=1  \ny: abs(-1-5)=6 \u2192 6-7=-1 \u2192 abs=1 \u2192 1-8=-7 \u2192 abs=7 \u2192 7-2=5  \nTotal: 1 + 5 = 6 \u2705  \n\nA passes all.\n\nNow check if others fail.\n\nTest B:  \nx=-15, y=-16  \nx: abs(-15+4)=11 \u2192 11-9=2 \u2192 2-2=0 \u2192 0-4=-4 \u2192 abs=4  \ny: same as A \u2192 4  \nTotal: 4+4=8 \u2705\n\nx=13, y=-3  \nx: abs(13+4)=17 \u2192 17-9=8 \u2192 8-2=6 \u2192 6-4=2 \u2192 abs=2  \ny: abs(-3-5)=8 \u2192 8-7=1 \u2192 1-8=-7 \u2192 abs=7 \u2192 7-2=5  \nTotal: 2 + 5 = 7 \u2705\n\nx=0, y=12  \nx: abs(0+4)=4 \u2192 4-9=-5 \u2192 abs=5 \u2192 5-2=3 \u2192 3-4=-1 \u2192 abs=1  \ny: abs(12-5)=7 \u2192 7-7=0 \u2192 0-8=-8 \u2192 abs=8 \u2192 8-2=6  \nTotal: 1+6=7 \u2705\n\nx=-12, y=-1  \nx: abs(-12+4)=8 \u2192 8-9=-1 \u2192 abs=1 \u2192 1-2=-1 \u2192 abs=1 \u2192 1-4=-3 \u2192 abs=3  \ny: same as A \u2192 5  \nTotal: 3+5=8 \u274c (should be 6) \u2192 fails\n\nB fails at x=-12, y=-1 \u2192 gives 8, expected 6 \u2192 reject.\n\nTest C:  \nx=-15, y=-16  \nx: abs(-15-4)=19 \u2192 19-9=10 \u2192 10-2=8 \u2192 8-4=4  \ny: abs(-16+5)=11 \u2192 11-7=4 \u2192 4-8=-4 \u2192 abs=4 \u2192 4-2=2  \nTotal: 4 + 2 = 6 \u274c (expected 8) \u2192 fail\n\nC fails first example \u2192 reject.\n\nTest D:  \nx=-15, y=-16  \nx: abs(-15+4)=11 \u2192 11-9=2 \u2192 2-2=0 \u2192 0-4=-4 \u2192 abs=4  \ny: abs(-16+5)=11 \u2192 11-7=4 \u2192 4-8=-4 \u2192 abs=4 \u2192 4-2=2  \nTotal: 4 + 2 = 6 \u274c (expected 8) \u2192 fail\n\nOnly A passes all.\n\nThe correct repair is A."
      },
      "penalty": {
        "answer": "A",
        "success": 1,
        "cost": 2049,
        "scratch_budget_hit": true,
        "scratch": "x=-15, y=-16 \u2192 8  \nx=13, y=-3 \u2192 7  \nx=0, y=12 \u2192 7  \nx=-12, y=-1 \u2192 6\n\nA. abs(abs(abs(abs(x - 4) - 9) - 2) - 4) + abs(abs(abs(abs(y - 5) - 7) - 8) - 2)  \nx=-15: abs(-15-4)=29 \u2192 abs(29-9)=20 \u2192 abs(20-2)=18 \u2192 abs(18-4)=14  \ny=-16: abs(-16-5)=21 \u2192 abs(21-7)=14 \u2192 abs(14-8)=6 \u2192 abs(6-2)=4  \nSum: 14+4=18 \u2192 not 8 \u2192 fail\n\nB. abs(abs(abs(abs(x + 4) - 9) - 2) - 4) + abs(abs(abs(abs(y - 5) - 7) - 8) - 2)  \nx=-15: -15+4=-11 \u2192 abs(-11)=11 \u2192 11-9=2 \u2192 2-2=0 \u2192 0-4=-4 \u2192 abs=4  \ny=-16: -16-5=-21 \u2192 abs=21 \u2192 21-7=14 \u2192 14-8=6 \u2192 6-2=4 \u2192 abs=4  \nSum: 4+4=8 \u2192 good  \nx=13: 13+4=17 \u2192 17-9=8 \u2192 8-2=6 \u2192 6-4=2 \u2192 abs=2  \ny=-3: -3-5=-8 \u2192 8 \u2192 8-7=1 \u2192 1-8=-7 \u2192 abs=7 \u2192 7-2=5 \u2192 abs=5  \nSum: 2+5=7 \u2192 good  \nx=0: 0+4=4 \u2192 4-9=-5 \u2192 abs=5 \u2192 5-2=3 \u2192 3-4=-1 \u2192 abs=1  \ny=12: 12-5=7 \u2192 7-7=0 \u2192 0-8=-8 \u2192 abs=8 \u2192 8-2=6 \u2192 abs=6  \nSum: 1+6=7 \u2192 good  \nx=-12: -12+4=-8 \u2192 8 \u2192 8-9=-1 \u2192 abs=1 \u2192 1-2=-1 \u2192 abs=1 \u2192 1-4=-3 \u2192 abs=3  \ny=-1: -1-5=-6 \u2192 6 \u2192 6-7=-1 \u2192 abs=1 \u2192 1-8=-7 \u2192 abs=7 \u2192 7-2=5 \u2192 abs=5  \nSum: 3+5=8 \u2192 should be 6 \u2192 fail\n\nC. abs(abs(abs(abs(x - 4) - 9) - 2) - 4) + abs(abs(abs(abs(y + 5) - 7) - 8) - 2)  \nx=-15: -15-4=-19 \u2192 19 \u2192 19-9=10 \u2192 10-2=8 \u2192 8-4=4 \u2192 abs=4  \ny=-16: -16+5=-11 \u2192 11 \u2192 11-7=4 \u2192 4-8=-4 \u2192 abs=4 \u2192 4-2=2 \u2192 abs=2  \nSum: 4+2=6 \u2192 not 8 \u2192 fail\n\nD. abs(abs(abs(abs(x + 4) - 9) - 2) - 4) + abs(abs(abs(abs(y + 5) - 7) - 8) - 2)  \nx=-15: -15+4=-11 \u2192 11 \u2192 11-9=2 \u2192 2-2=0 \u2192 0-4=-4 \u2192 abs=4  \ny=-16: -16+5=-11 \u2192 11 \u2192 11-7=4 \u2192 4-8=-4 \u2192 abs=4 \u2192 4-2=2 \u2192 abs=2  \nSum: 4+2=6 \u2192 not 8 \u2192 fail\n\nOnly B passes first three, fails last.  \nWait: last example: x=-12, y=-1 \u2192 should be 6  \nB: x=-12: -12+4=-8 \u2192 8 \u2192 8-9=-1 \u2192 abs=1 \u2192 1-2=-1 \u2192 abs=1 \u2192 1-4=-3 \u2192 abs=3  \ny=-1: -1+5=4 \u2192 4-7=-3 \u2192 abs=3 \u2192 3-8=-5 \u2192 abs=5 \u2192 5-2=3 \u2192 abs=3  \nSum: 3+3=6 \u2192 good\n\nRecheck all:\n\nB:  \nx=-15, y=-16 \u2192 4+4=8  \nx=13, y=-3 \u2192 x:17\u21928\u21926\u21922\u21922 (wait: 17-9=8; 8-2=6; 6-4=2) \u2192 2  \ny=-3: -3-5=-8\u21928\u21928-7=1\u21921-8=-7\u21927\u21927-2=5\u21925 \u2192 sum=2+5=7  \nx=0: 0+4=4\u21924-9=-5\u21925\u21925-2=3\u21923-4=-1\u21921 \u2192 y=12:12+5=17\u219217-7=10\u219210-8=2\u21922-2=0 \u2192 sum=1+0=1 \u2192 should be 7 \u2192 fail!\n\nWait: y=12: in B, y term is abs(abs(abs(abs(y-5)-7)-8)-2)  \ny=12: 12-5=7 \u2192 7-7=0 \u2192 0-8=-8 \u2192 abs=8 \u2192 8-2=6 \u2192 abs=6  \nx=0: x+4=4 \u2192 4-9=-5 \u2192 5 \u2192 5-2=3 \u2192 3-4=-1 \u2192 abs=1  \nSum: 1+6=7 \u2192 good\n\nx=-12: x+4=-8 \u2192 8 \u2192 8-9=-1 \u2192 1 \u2192 1-2=-1 \u2192 1 \u2192 1-4=-3 \u2192 3  \ny=-1: y-5=-6 \u2192 6 \u2192 6-7=-1 \u2192 1 \u2192 1-8=-7 \u2192 7 \u2192 7-2=5 \u2192 5  \nSum: 3+5=8 \u2192 should be 6 \u2192 fail\n\nWait: in B, y term is abs(abs(abs(abs(y-5)-7)-8)-2) \u2192 y=-1 \u2192 -1-5=-6 \u2192 6 \u2192 6-7=-1 \u2192 1 \u2192 1-8=-7 \u2192 7 \u2192 7-2=5 \u2192 5  \nx=-12: x+4=-8 \u2192 8 \u2192 8-9=-1 \u2192 1 \u2192 1-2=-1 \u2192 1 \u2192 1-4=-3 \u2192 3  \n3+5=8 \u2260 6 \u2192 fail\n\nSo B fails last.\n\nTry C again:\n\nC: abs(abs(abs(abs(x-4)-9)-2)-4) + abs(abs(abs(abs(y+5)-7)-8)-2)\n\nx=-15: -15-4=-19 \u219219\u219219-9=10\u21928\u21928-2=6\u21926-4=2 \u2192 2  \ny=-16: -16+5=-11\u219211\u219211-7=4\u21924-8=-4\u21924\u21924-2=2 \u2192 2  \nSum: 2+2=4 \u2192 not 8\n\nD: x+4, y+5  \nx=-15: -11\u219211\u21922\u21920\u21924  \ny=-16: -11\u219211\u21924\u21924\u21922 \u2192 sum 4+2=6 \u2192 not 8\n\nNone seem to work.\n\nWait \u2014 perhaps a typo in reasoning.\n\nGo back to B:  \nx=-12, y=-1 \u2192 output must be 6\n\nx=-12: x+4 = -8 \u2192 | -8 | = 8  \n8 - 9 = -1 \u2192 | -1 | = 1  \n1 - 2 = -1 \u2192 | -1 | = 1  \n1 - 4 = -3 \u2192 | -3 | = 3\n\ny=-1: y-5 = -6 \u2192 6  \n6 - 7 = -1 \u2192 1  \n1 - 8 = -7 \u2192 7  \n7 - 2 = 5 \u2192 5\n\n3+5=8 \u2192 not 6\n\nSo fails.\n\nTry C: y term uses y+5 \u2192 y=-1 \u2192 4 \u2192 4-7=-3 \u2192 3 \u2192 3-8=-5 \u21925 \u21925-2=3 \u21923  \nx term: x-4 = -16-4"
      }
    }
  ],
  "example_rule": "First (task_id, sample, seed) in each nonempty control/penalty success cell; four illustrations, not representative effect estimates.",
  "limitations": [
    "One training seed and reused inspected development tasks.",
    "Control differs in deterministic runtime; later trajectories are not shared between policies.",
    "Same-row reward rescoring reconstructs estimator support, not a trained counterfactual.",
    "Clipping acts before Adam; norm or update count cannot establish the cause of the endpoint.",
    "No intermediate checkpoint evaluation, new hypotheses or inferential tests are retroactively registered."
  ]
}