
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 $P(x)$ be a polynomial defined as $P(x) = 3x^5 - 2x^3 + 5x^2 - 4x + 1$. A new polynomial $Q(x)$ is formed by replacing each nonzero coefficient of $P(x)$ with their mean. Evaluate $P(1)$ and $Q(1)$ and check if they are equal.

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