
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 a circle $O: x^{2}+y^{2}=9$, where point $A$ lies on circle $O$, and the point $C$ is the reflection of point $A$ with respect to the $x$-axis. Point $C$ lies on the line passing through point $D(\frac{9}{4},0)$, and the line passing through points $C$ and $D$ intersects circle $O$ at another point $B$. The line $AB$ intersects the $x$-axis at point $T$.  
$(1)$ Find the equation of the locus of the midpoint $P$ of $AD$.  
$(2)$ Find the maximum area of triangle $TBC$.

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