
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 $\triangle ABC$, the coordinates of point $A$ are $(\sqrt{3}, 3)$, the equation of the median $CM$ on side $AB$ is $5\sqrt{3}x + 9y - 18 = 0$, and the equation of the angle bisector $BT$ on $\angle B$ is $y = 1$. Find:

1. The coordinates of vertex $B$;
2. The area of $\triangle ABC$.

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