
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
Given the function $f(x) = 2\sqrt{3}\sin x \sin\left(\frac{\pi}{2}-x\right) - 2\cos(\pi+x)\cos x + 2$.
1. Find the smallest positive period of $f(x)$.
2. In triangle $ABC$, let $a$, $b$, and $c$ be the sides opposite angles $A$, $B$, and $C$, respectively. If $f(A) = 4$, $b = 1$, and the area of triangle $ABC$ is $\frac{\sqrt{3}}{2}$, find the value of $a$.

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