
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 the following conditions:①$\left(2b-c\right)\cos A=a\cos C$, ②$a\sin\ \ B=\sqrt{3}b\cos A$, ③$a\cos C+\sqrt{3}c\sin A=b+c$, choose one of these three conditions and complete the solution below.<br/>Question: In triangle $\triangle ABC$, with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively, satisfying ______, and $c=4$, $b=3$.<br/>$(1)$ Find the area of $\triangle ABC$;<br/>$(2)$ If $D$ is the midpoint of $BC$, find the cosine value of $\angle ADC$.<br/>Note: If multiple conditions are chosen and answered separately, the first answer will be scored.

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