
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
Please choose one of the following three conditions to fill in the blank and answer the question.<br/>①$2a\cos B=b+2c$；②$2\sin C+\tan A\cos B+\sin B=0$；③$\frac{a-c}{\sin B}=\frac{b+c}{\sin A+\sin C}$.<br/>Given that in triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, with $b=2$ and $c=4$, and it satisfies ____.<br/>$(1)$ Find $A$;<br/>$(2)$ If point $D$ lies on side $BC$ and $BC=4BD$, find the length of $AD$.

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