
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 $\triangle ABC$, let the sides opposite to angles $A$, $B$, and $C$ be $a$, $b$, and $c$, respectively. The following three conditions are satisfied: $①a^{2}+b^{2}=c^{2}+ab$; $② \sqrt {3}c=14\sin C$; $③a+b=13$.

Find:
$(1)$ The measure of internal angle $C$ and the length of side $c$.
$(2)$ The area of $\triangle ABC$.

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