
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 vectors $\overrightarrow{m}=(\cos x,\sin x)$, $\overrightarrow{n}=(2 \sqrt {2}+\sin x,2 \sqrt {2}-\cos x)$, and the function $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$, where $x\in R$.
1. Find the maximum value of function $f(x)$.
2. If $x\in(-\frac{3}{2}\pi,-\pi)$ and $f(x)=1$, find the value of $\cos (x+\frac{5}{12}\pi)$.

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