
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 $a=(\sqrt{3}\sin x, \sin x)$ and $b=(\cos x, \sin x)$, where $x \in \left[0, \frac{\pi}{2}\right]$.

(1) Find the value of $x$ when $|a|=|b|$;

(2) Define the function $f(x) = a \cdot b$, find the maximum value of $f(x)$.

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