
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 $\sin \alpha = \frac {4 \sqrt {3}}{7}, \cos (\beta - \alpha) = \frac {13}{14}$, and $0 < \beta < \alpha < \frac {\pi}{2}$.
$(1)$ Find the value of $\tan 2\alpha$;
$(2)$ Find the value of $\cos \beta$.

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