
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 $f(x)=2\sqrt{3}\sin x\cos x+2\cos^2x.$  
$(1)$ Find the value of $f\left( \dfrac{\pi}{24}\right)$.  
$(2)$ If the function $f(x)$ is monotonically increasing in the interval $[-m,m]$, find the maximum value of the real number $m$.

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