
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
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. Let the plane vectors $\overrightarrow {e_{1}} = (2\cos C, \frac {c}{2} - b)$ and $\overrightarrow {e_{2}} = (\frac {1}{2}a, 1)$, and $\overrightarrow {e_{1}}$ is perpendicular to $\overrightarrow {e_{2}}$.
(I) Find the value of $\cos 2A$;
(II) If $a=2$, find the range of the perimeter $L$ of $\triangle ABC$.

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