
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 acute triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\left(a-b+c\right)\left(a-b-c\right)+ab=0$ and $bc\sin C=3c\cos A+3a\cos C$. Find:<br/>
$(1)$ the value of $c$;<br/>
$(2)$ the range of values for $a+b$.

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