
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 angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given _____.<br/>①$c\sin B+2\cos A=b\sin C+1$; ②$\cos 2A-3\cos \left(B+C\right)-1=0$; ③Vector $\overrightarrow{m}=(\sqrt{3}b,a)$, vector $\overrightarrow{n}=\left(\sin B,\cos A\right)$, and $\overrightarrow{m}∥\overrightarrow{n}$. Choose one of these three conditions to fill in the blank line and answer the following:<br/>$(1)$ Find the measure of angle $A$;<br/>$(2)$ If the area of $\triangle ABC$ is $\frac{\sqrt{3}}{2}$, find the minimum value of $a$.

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