
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 $\Delta ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively, and $\sqrt{3}a\sin B - b\cos A = b$,

$(1)$ Find the size of $\angle A$;

$(2)$ If $b+c=4$, when $a$ takes the minimum value, find the area of $\Delta ABC$;

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