
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 $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to the internal angles $A$, $B$, and $C$ respectively. Given that $\cos(B-C)\cos A + \cos 2A = 1 + \cos A\cos(B+C)$. Find:<br/>
$(1)$ If $B=C$, determine the value of $\cos A$.<br/>
$(2)$ Find the value of $\frac{b^{2}+c^{2}}{a^{2}}$.

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