
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 that point $T$ is a moving point on the circle $A: \left(x-1\right)^{2}+y^{2}-8=0$, point $B(-1,0)$, and the perpendicular bisector of segment $BT$ intersects segment $AT$ at point $S$. Let the locus of point $S$ be curve $C$.  
$(1)$ Find the equation of curve $C$;  
$(2)$ Passing through $B(-1,0)$, draw two chords $DE$ and $MN$ on curve $C$, with midpoints $P$ and $Q$, respectively. If $\overrightarrow{DE} \cdot \overrightarrow{MN} = 0$, find the maximum area of triangle $BPQ$.

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