
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 the functions $f(x)=\cos^{2}x$ and $g(x)=1+\frac{1}{2}\sin 2x$.

(1) Let $x=x_{0}$ be a symmetry axis of the graph of the function $y=f(x)$. Find the value of $g(2x_{0})$.

(2) Find the range of the function $h(x)=f(x)+g(x)$, where $x\in\left[0,\frac{\pi}{4}\right]$.

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