
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 function $f(x)=\sin(\pi-x)\sin(\frac{\pi}{2}-x)+\cos^2x$.  
$(1)$ Find the smallest positive period and axis of symmetry of the function $f(x)$.  
$(2)$ When $x\in[-\frac{\pi}{8},\frac{3\pi}{8}]$, find the interval where the function $f(x)$ is monotonically increasing.

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