
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 $f(x)=\frac{\sin 2x + 2\cos^2x}{\cos x}$.

$(1)$ Find the domain of $f(x)$ and the value of $f\left( \frac{\pi}{4} \right)$;

$(2)$ Find the interval of monotonic increase for $f(x)$ in $\left( 0, \frac{\pi}{2} \right)$.

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