
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 $\triangle ABC$, $a$, $b$, and $c$ represent the sides opposite to angles $A$, $B$, and $C$ respectively. Given that $\cos^2{A} - \cos^2{B} + \sin^2{C} = \sin{B}\sin{C} = \frac{1}{4}$, and the area of $\triangle ABC$ is $\sqrt{3}$, find the value of $a$.

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