
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
Given the septenary polynomial
\[ 
Q\left(x_{1}, x_{2}, \cdots, x_{7}\right) =\left(x_{1}+x_{2}+\cdots+x_{7}\right)^{2}+2\left(x_{1}^{2}+x_{2}^{2}+\cdots+x_{7}^{2}\right),
\]
which can be expressed as the sum of squares of polynomials with non-negative integer coefficients, that is
\[
Q\left(x_{1}, x_{2}, \cdots, x_{7}\right) = P_{1}^{2}\left(x_{1}, x_{2}, \cdots, x_{7}\right) + P_{2}^{2}\left(x_{1}, x_{2}, \cdots, x_{7}\right) + \cdots + P_{7}^{2}\left(x_{1}, x_{2}, \cdots, x_{7}\right),
\]
where \(P_{k}\left(x_{1}, x_{2}, \cdots, x_{7}\right)\) are polynomials with non-negative integer coefficients. Find all possible values of \(P_{1}(1,1, \cdots, 1)\).

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