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

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