
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 vectors $\overrightarrow{a} = (\sqrt{3}\sin x, \sin x)$ and $\overrightarrow{b} = (\cos x, \sin x)$.
(1) If $|\overrightarrow{a}| = |\overrightarrow{b}|$ and $x \in [0, \frac{\pi}{2}]$, find the value of $x$;
(2) Let the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b}$; determine the interval where $f(x)$ is monotonically increasing.

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