
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 \( P(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n \) be a polynomial in \( x \) where the coefficients \( a_0, a_1, a_2, \ldots, a_n \) are non-negative integers. If \( P(1) = 25 \) and \( P(27) = 1771769 \), find the value of \( a_0 + 2a_1 + 3a_2 + \cdots + (n+1)a_n \).

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