### SYSTEM: interrogating function with args ['integer', 'integer', 'integer']

--- LLM ---
  None

### SYSTEM: calling tool
  (1, 2, 3) → "5"

--- LLM ---
  None

### SYSTEM: calling tool
  (0, 0, 0) → "3"

--- LLM ---
  None

### SYSTEM: calling tool
  (2, 4, 6) → "9"

--- LLM ---
  None

### SYSTEM: calling tool
  (3, 3, 3) → "6"

--- LLM ---
  None

### SYSTEM: calling tool
  (5, 2, 8) → "12"

--- LLM ---
  None

### SYSTEM: calling tool
  (4, 1, 7) → "9"

--- LLM ---
  None

### SYSTEM: calling tool
  (10, 10, 10) → "27"

--- LLM ---
  None

### SYSTEM: calling tool
  (2, 6, 4) → "9"

--- LLM ---
  None

### SYSTEM: calling tool
  (7, 0, 0) → "6"

--- LLM ---
  None

### SYSTEM: calling tool
  (0, 5, 0) → "6"

--- LLM ---
  None

### SYSTEM: calling tool
  (0, 0, 5) → "8"

--- LLM ---
  None

### SYSTEM: calling tool
  (6, 2, 4) → "9"

--- LLM ---
  None

### SYSTEM: calling tool
  (5, 0, 0) → "4"

--- LLM ---
  None

### SYSTEM: calling tool
  (1, 0, 0) → "2"

--- LLM ---
  None

### SYSTEM: calling tool
  (0, 7, 0) → "8"

--- LLM ---
  None

### SYSTEM: calling tool
  (0, 1, 0) → "2"

--- LLM ---
  None

### SYSTEM: calling tool
  (0, 0, 1) → "4"

--- LLM ---
  None

### SYSTEM: calling tool
  (2, 0, 0) → "1"

--- LLM ---
  None

### SYSTEM: calling tool
  (3, 0, 0) → "2"

--- LLM ---
  None

### SYSTEM: calling tool
  (0, 2, 0) → "3"

--- LLM ---
  After systematically probing the mystery function I found a concise rule that reproduces every one of the 20 test-
  results exactly.

  Notation
  Let I(cond) = 1 if cond is true, otherwise 0.
  Assume the three inputs a, b, c are non-negative integers (all our tests were non-negative).

  Rule

  1.  Start with the ordinary sum of the three inputs and add 3:

          result = a + b + c + 3

  2.  Apply three fixed “penalties” (subtract 2 each time a condition is met)

          result -= 2 · I(b > 0)        // subtract 2 if b is positive
          result -= 2 · I(a > 0)        // subtract 2 if a is positive
          result -= 2 · I(a > 1)        // subtract another 2 if a is greater than 1

     (When a ≥ 2 both a>0 and a>1 are true, so a positive value ≥2 costs a total of 4.
      If a = 1 it costs 2, and if a = 0 it costs nothing.
      The variable c never invokes a penalty.)

  That is all the function does.

  Equivalent piece-wise description

  •  Base = a + b + c + 3
  •  If b ≠ 0  →  subtract 2
  •  If a = 1  →  subtract 2
  •  If a ≥ 2  →  subtract 4

  Examples (matches every call we made)
    (1,2,3): 1+2+3+3 = 9 −2(b) −2(a=1)   = 5
    (0,0,0): 0+0+0+3 = 3                      = 3
    (5,2,8): 5+2+8+3 = 18 −2(b) −4(a≥2)   = 12
    (0,5,0): 0+5+0+3 = 8 −2(b)              = 6
    (0,0,5): 0+0+5+3 = 8                    = 8

  So the mystery function is simply:

      return a + b + c + 3
             - 2*(b > 0)
             - 2*(a > 0)
             - 2*(a > 1);

  (or any equivalent formulation).

### SYSTEM: The tool was used 20 times.

### SYSTEM: verifying function with args ['integer', 'integer', 'integer']

### SYSTEM: inputs:
  {'a': 2, 'b': 1, 'c': 0}

--- LLM ---
  Using the rule I inferred: result = a + b + c + 3 - 2*(b>0) - 2*(a>0) - 2*(a>1).
  For a=2, b=1, c=0:
  1. Base: 2+1+0+3 = 6
  2. Penalties:  b>0 -> -2 (4)
                 a>0 -> -2 (2)
                 a>1 -> -2 (0)
  Expected output is therefore 0.

  `0`

### SYSTEM: WRONG