
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
Given the function \( f(x) = a x^{3} + b x^{2} + c x + d \) (where \( a \neq 0 \)), and knowing that \( \left| f^{\prime}(x) \right| \leq 1 \) when \( 0 \leq x \leq 1 \), find the maximum value of \( a \).

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