
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. The vectors $\overrightarrow{m}=\left(c,a-b\right)$ and $\overrightarrow{n}=\left(\sin B-\sin C,\sin A+\sin B\right)$ are given, and $\overrightarrow{m}\bot \overrightarrow{n}$. <br/>$(1)$ Find the measure of angle $A$; <br/>$(2)$ If $D$ is a point on $AC$ such that $AD=BD$ and $BC=3$, find the maximum area of triangle $BCD$.

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