
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 $a$, $b$, $c$ are opposite to the angles $A$, $B$, $C$ respectively, and it is given that $\sqrt{3}c \cos A + a \sin C = \sqrt{3}c$.
$(1)$ Find the magnitude of angle $A$.
$(2)$ If $b + c = 5$ and the area of triangle $ABC$ is $S_{\triangle ABC} = \sqrt{3}$, find the value of $a$.

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