
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 $\overrightarrow{m}=(2\cos (x+ \frac{\pi}{2}),\cos x)$ and $\overrightarrow{n}=(\cos x,2\sin (x+ \frac{\pi}{2}))$, and the function $f(x)= \overrightarrow{m} \cdot \overrightarrow{n}+1$.
(1) Assuming the equation $f(x)-1=0$ has two zero points $x_{1}$, $x_{2}$ in $(0,\pi)$, find the value of $x_{1}+x_{2}$;
(2) If the graph of the function $y=f(x)$ is translated to the left by $\frac{\pi}{6}$ units, and then translated downwards by $2$ units to obtain the graph of function $g(x)$, find the intervals in which $g(x)$ is monotonically increasing in $\left[- \frac{\pi}{2}, \frac{\pi}{2}\right]$.

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