
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 an acute triangle $\triangle ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively. Choose one of the conditions from condition 1 and condition 2 as known, find:<br/>$(1)$ The degree of angle $A$<br/>$(2)$ If $c=1$, find the range of values for the area of $\triangle ABC$.<br/>Condition 1: $a\cos C+\sqrt{3}a\sin C=b+c$;<br/>Condition 2: The area of $\triangle ABC$ is $S=\frac{\sqrt{3}({b}^{2}+{c}^{2}-{a}^{2})}{4}$.

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