
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

(a) Determine \(a\), \(b\), and \(c\) such that the equation

\[
(n+2)^{2} = a(n+1)^{2} + b n^{2} + c(n-1)^{2}
\]

is true for any number \(n\).

(b) Suppose \(x_{1}, x_{2}, \ldots, x_{7}\) satisfy the system

\[
\left\{
\begin{array}{l}
x_{1} + 4 x_{2} + 9 x_{3} + 16 x_{4} + 25 x_{5} + 36 x_{6} + 49 x_{7} = 1 \\
4 x_{1} + 9 x_{2} + 16 x_{3} + 25 x_{4} + 36 x_{5} + 49 x_{6} + 64 x_{7} = 12 \\
9 x_{1} + 16 x_{2} + 25 x_{3} + 36 x_{4} + 49 x_{5} + 64 x_{6} + 81 x_{7} = 123
\end{array}
\right.
\]

Determine the value of

\[
16 x_{1} + 25 x_{2} + 36 x_{3} + 49 x_{4} + 64 x_{5} + 81 x_{6} + 100 x_{7}
\]

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