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

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