
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 the vectors \(\boldsymbol{a}=\left(2 \cos \frac{x}{2}, \tan \left(\frac{x}{2}+\frac{\pi}{4}\right)\right)\) and \(\boldsymbol{b}=\left(\sqrt{2} \sin \left(\frac{x}{2}+\frac{\pi}{4}\right), \tan \left(\frac{x}{2}-\frac{\pi}{4}\right)\right)\), let \(f(x)=\boldsymbol{a} \cdot \boldsymbol{b}\). Find the maximum value of the function \(f(x)\), the smallest positive period, and determine the intervals of monotonicity of \(f(x)\) on \(\left[0, \frac{\pi}{2}\right]\).

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