
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 $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$, respectively. It is known that $2a\cos A = c\cos B + b\cos C$.
(Ⅰ) Find the value of $\cos A$;
(Ⅱ) If $a=1$ and $\cos^2 \frac{B}{2} + \cos^2 \frac{C}{2} = 1+ \frac{\sqrt{3}}{4}$, find the value of side $c$.

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