
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 $\overrightarrow{a}=(\sin x,1)$ and $\overrightarrow{b}=(\sin x,\cos x+1)$,
(I) Find the coordinates of all vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that are parallel.
(II) If the function $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}$, $x\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$, find the maximum value of the function $f(x)$ and the corresponding $x$ value.

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