
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 $f(x)=\frac{1}{2}{\cos^4}x+\sqrt{3}\sin{x}\cos{x}-\frac{1}{2}{\sin^4}x+m$ has a maximum value of $\frac{3}{2}$,  
$(1)$ Find the value of the constant $m$ and determine the corresponding set of $x$ values when $f(x)$ attains its maximum value;  
$(2)$ Find the intervals where $f(x)$ is monotonically increasing.

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