
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 $cosα=\frac{1}{7}, α∈({-\frac{π}{2},0})$.  
$(1)$ Find the value of $cos({\frac{π}{3}-α})$;  
$(2)$ If $sin({α+β})=-\frac{{3\sqrt{3}}}{{14}}, β∈({0,\frac{π}{2}})$, find the value of $\beta$.

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