
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 $\overrightarrow{a}=(\sin x, \sqrt {3}\sin x)$ and $\overrightarrow{b}=(\sin x,\cos x)$, where $x\in[0, \dfrac {\pi}{2}]$.
(I) Find the value of $x$ when $|\overrightarrow{a}|=|\overrightarrow{b}|$;
(II) Define a function $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}$. By shifting the graph of $f(x)$ to the left by $\dfrac {\pi}{6}$ units, we obtain the graph of function $g(x)$. Find the maximum value of $g(x)$ and the corresponding value of $x$.

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