
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
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are respectively $a$, $b$, and $c$, with $a+\frac{1}{a}=4\cos C$ and $b=1$.

$(1)$ If $\sin C=\frac{\sqrt{21}}{7}$, find $a$ and $c$;

$(2)$ If $\triangle ABC$ is a right-angled triangle, find the area of $\triangle ABC$.

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