
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 real numbers \(a\) and \(b\) satisfy:

\[
\begin{aligned}
a^{2}\left(b^{2}+1\right) + b(b + 2a) &= 40, \\
a(b + 1) + b &= 8.
\end{aligned}
\]

Find the value of \(\frac{1}{a^{2}}+\frac{1}{b^{2}}\).

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