
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 $\overrightarrow{a}=( \sqrt {3}\sin 2x,\cos 2x)$ and $\overrightarrow{b}=(\cos 2x,-\cos 2x)$,
(I) If $x\in( \frac {7\pi}{24}, \frac {5\pi}{12})$, and $\overrightarrow{a}\cdot \overrightarrow{b}+ \frac {1}{2}=- \frac {3}{5}$, find $\cos 4x$;
(II) If $x\in(0, \frac {\pi}{3}]$ and the equation $\overrightarrow{a}\cdot \overrightarrow{b}+ \frac {1}{2}=m$ has exactly one real root, find the value of $m$.

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