
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 \( f(x) = a x - \frac{3}{2} x^2 \) has a maximum value of no more than \( \frac{1}{6} \), and when \( x \in \left[ \frac{1}{4}, \frac{1}{2} \right] \), \( f(x) \geqslant \frac{1}{8} \), find the value of \( a \).

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