
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 $f\left( x \right)=\left( m+2{{\cos }^{2}}x \right)\cdot \cos \left( 2x+\theta \right)$ is an odd function, and $f\left( \frac{\pi }{4} \right)=0$, where $m\in R,\theta \in \left( 0,\pi \right)$.

(I) Find the center of symmetry and the monotonically increasing interval of the function $f\left( x \right)$;

(II) In $\Delta ABC$, angles $A,B,C$ have opposite sides $a,b,c$ respectively, and $f\left( \frac{C}{2}+\frac{\pi }{24} \right)=-\frac{1}{2}$, $c=1,ab=2\sqrt{3}$, find the perimeter of $\Delta ABC$.

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