
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 in $\triangle ABC$, $2B=A+C$, and $c=2a$.
$(1)$ Find the sizes of angles $A$, $B$, $C$;
$(2)$ Let the sequence $\{a_n\}$ satisfy $a_n=2^n|\cos nC|$, and the sum of the first $n$ terms be $S_n$. If $S_n=20$, find the value of $n$.

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