
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 acute triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Let $S$ be the area of $\triangle ABC$, and it is given that $2S=a^{2}-\left(b-c\right)^{2}$,<br/>$(1)$ Find $\cos A$;<br/>$(2)$ If $a=\frac{\sqrt{41}}{2}$ and $S=4$, find the perimeter of $\triangle ABC$.

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