
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  $F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|$ .  When  $a$ ,  $b$ ,  $c$  run over all the real numbers, find the smallest possible value of  $F$ .

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