
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 the function \( f(x) = x \sin x \) (with \( x \in \mathbf{R} \)) achieve an extremum at \( x = x_{0} \). Determine the value of \( \left(1 + x_{0}^{2}\right)\left(1 + \cos 2 x_{0}\right) = \quad . \)

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