
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
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that the radius of the circumcircle of $\triangle ABC$ is $1$, and $b\sin B + c\sin C = \left(\frac{2\sqrt{3}}{3}b\sin C + a\right)\sin A$.  
$(1)$ Find angle $A$;  
$(2)$ If $AC = \sqrt{2}$, $AD$ is the angle bisector of $\triangle ABC$, find the length of $AD$.

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