
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 $α∈({0,\frac{π}{4}})$, $β∈({\frac{{3π}}{4},π})$, and $\sin(α-β)=-\frac{3}{5}$.  
$(1)$ If $\sinβ=\frac{5}{{13}}$, find the value of $\cos \alpha$;  
$(2)$ If $\frac{{\tanβ}}{{\tanα}}=-11$, find the value of $\alpha +\beta$.

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