
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 $f(x)=2\sqrt{3}\sin x \cos x + 2\cos^2 x + 1$.
$(1)$ Find the interval where the function $f(x)$ is monotonically increasing.
$(2)$ If the inequality $f(x) > m - 3$ holds for any $x \in \left(\frac{\pi}{4}, \pi\right)$, find the range of real number $m$.

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