
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 $M(1,\frac{3}{2})$ lies on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 (a>b>0)$, where $A$ and $B$ are the left and right vertices of the ellipse, the sum of the slopes of lines $MA$ and $MB$ satisfies: $k_{MA}+k_{MB}=-1$.  
$(1)$ Find the standard equation of the ellipse;  
$(2)$ If a line with slope $1$ intersects the ellipse at points $P$ and $Q$, does there exist a fixed point $T$ on the ellipse such that the sum of the slopes of lines $PT$ and $QT$ satisfies $k_{PT}+k_{QT}=0$ (where $P$, $Q$, and $T$ are distinct points)? If so, find the coordinates of point $T$; if not, explain the reason.

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