
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
Please choose one of the following conditions:①$asinB-\sqrt{3}bcosBcosC=\sqrt{3}cco{s}^{2}B$；②$bcosC+\frac{1}{2}c=a$；③$\frac{\sqrt{3}bsinA}{1+cosB}=a$, and supplement it into the following question, then solve it. (If no choice is made, it will be scored according to choice ①). In $\triangle ABC$, where $a$, $b$, $c$ are the side lengths opposite to angles $A$, $B$, $C$ respectively, if ________.<br/>$(1)$ Find the measure of angle $B$;<br/>$(2)$ If $\triangle ABC$ is an acute triangle and $c=1$, find the range of values for $a^{2}+b^{2}$.

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