
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
Given vectors \(\overrightarrow{O P}=\left(2 \cos \left(\frac{\pi}{2}+x\right),-1\right)\) and \(\overrightarrow{O Q}=\left(-\sin \left(\frac{\pi}{2}- x\right), \cos 2 x\right)\), and the function \(f(x)=\overrightarrow{O P} \cdot \overrightarrow{O Q}\). If \(a, b, c\) are the sides opposite angles \(A, B, C\) respectively in an acute triangle \( \triangle ABC \), and it is given that \( f(A) = 1 \), \( b + c = 5 + 3 \sqrt{2} \), and \( a = \sqrt{13} \), find the area \( S \) of \( \triangle ABC \).

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