
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 \( x_{1}, x_{2}, \cdots, x_{7} \) be real numbers that satisfy the following equations:
\[
\begin{array}{l}
\sum_{k=1}^{7} k^{2} x_{k}=1, \\
\sum_{k=1}^{7}(k+1)^{2} x_{k}=12, \\
\sum_{k=1}^{7}(k+2)^{2} x_{k}=123,
\end{array}
\]
Find the value of \( \sum_{k=1}^{7}(k+3)^{2} x_{k} \).

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