
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 that $a$, $b$, $c$ are the opposite sides of angles $A$, $B$, $C$ in triangle $\triangle ABC$, and $b=5$, $c=7$. If $\triangle ABC$ satisfies one of the following two conditions: ① $\frac{a+c}{b}=\frac{sinB+sinA}{sinC-sinA}$; ② $cos^{2}(\frac{B-A}{2})-sinBsinA=\frac{3}{4}$. Please choose one condition from ① or ② to complete the following questions. I choose ____ (fill in ① or ②). <br/> $(1)$ Find $C$; <br/> $(2)$ Find the area of the corresponding $\triangle ABC$.

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