
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 $\triangle ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively, where $a=2$, $B=\frac{π}{3}$. Choose one of the following conditions ①, ②, ③ as the known condition to ensure the existence and uniqueness of $\triangle ABC$. <br/>$(1)$ Find the value of $c$; <br/>$(2)$ Find the area of $\triangle ABC$. <br/>Condition ①: $cosA=\frac{2\sqrt{7}}{7}$; Condition ②: $b=\frac{\sqrt{7}}{2}$; Condition ③: $b=\sqrt{7}$. <br/>Note: If the chosen condition does not meet the requirements, 0 points will be awarded; if multiple valid conditions are chosen and answered separately, the first answer will be scored.

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