
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 \( P(x) = x^{4} + a x^{3} + b x^{2} + c x + d \), where \( a, b, c, \) and \( d \) are real coefficients. Given that

\[ P(1) = 7, \quad P(2) = 52, \quad P(3) = 97, \]

find the value of \(\frac{P(9) + P(-5)}{4}\).

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