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

(Ⅰ) Find the range of $x$ for which the inequality $f(x)\geqslant \frac{1}{2}$ holds;

(Ⅱ) Let $\Delta ABC$ have internal angles $A$, $B$, $C$ with opposite sides $a$, $b$, $c$ respectively. If $f\left(\frac{B}{2}\right)=1$, $b=1$, and $c=\sqrt{3}$, find the value of $a$.

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