
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
Determine the value of $b$ for the polynomial $P(x) = x^3 + ax^2 + bx + c$ given that:
1. The mean of the zeros, the product of the zeros, and the sum of its coefficients ($1+a+b+c$) are all equal.
2. The y-intercept of the polynomial's graph is 5.

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