
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 \( x_1, x_2, \ldots, x_8 \) be real numbers such that:
\[\begin{aligned}
x_1 + 4x_2 + 9x_3 + 16x_4 + 25x_5 + 36x_6 + 49x_7 + 64x_8 &= 2, \\
9x_1 + 16x_2 + 25x_3 + 36x_4 + 49x_5 + 64x_6 + 81x_7 + 100x_8 &= 24, \\
16x_1 + 25x_2 + 36x_3 + 49x_4 + 64x_5 + 81x_6 + 100x_7 + 121x_8 &= 246, \\
25x_1 + 36x_2 + 49x_3 + 64x_4 + 81x_5 + 100x_6 + 121x_7 + 144x_8 &= 1234.
\end{aligned}\]
Find the value of \(36x_1 + 49x_2 + 64x_3 + 81x_4 + 100x_5 + 121x_6 + 144x_7 + 169x_8\).

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