
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
Consider the polynomial \( Q(x) = 3x^3 + dx^2 + ex + f \) which has the property that the mean of its zeros, the product of its zeros, and the sum of the coefficients are all equal. The \( y \)-intercept of the graph of \( y = Q(x) \) is 27. Determine the value of \( e \).

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