
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 vectors $\overrightarrow{m}=(\cos x, \sin x)$ and $\overrightarrow{b}=(-\cos x, \sqrt{3}\cos x)$, where $x \in \mathbb{R}$, and the function $f(x)=\overrightarrow{m} \cdot (\frac{1}{2} \overrightarrow{m}-\overrightarrow{n})$.

1. Find the range of $x$ for which the inequality $f(x) \geq \frac{1}{2}$ holds.
2. Let $a$, $b$, and $c$ be the sides opposite to angles $A$, $B$, and $C$ in $\triangle ABC$, respectively. If $f(\frac{B}{2})=1$, $b=1$, and $c=\sqrt{3}$, find the value of $a$.

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