
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
Choose one of the conditions $a+c=13$, $b=7$, $a+b+c=20$ to fill in the blank line below and complete the question. Given that in triangle $\triangle ABC$, the lengths of the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, $c$, and $c\cos A-2b\cos B+a\cos C=0$.
$(1)$ Find angle $B$;
$(2)$ If ____, $c \gt a$, $\overrightarrow{BA}•\overrightarrow{BC}=20$, find $\sin A$.

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