
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let \( A = (1, 0), B = (5, 4), \) and \( C = (p, q) \) be three points on the parabola \( y = -x^2 + 8x - 12 \), where \( 1 \le p \le 5 \). Find the largest possible area of triangle ABC.

Present the answer in LaTex format: \boxed{Your answer}
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