
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 an ellipse $M:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ passing through the points ($\frac{1}{2}$,$\frac{\sqrt{15}}{4}$) and (1,$\frac{\sqrt{3}}{2}$).<br/>$(1)$ Find the standard equation and eccentricity of the ellipse $M$.<br/>$(2)$ If the line $y=kx+3$ intersects the ellipse $M$ at points $A$ and $B$, does there exist a point $P$ on the $y$-axis such that the sum of the slopes of $PA$ and $PB$ is zero? If yes, find the coordinates of point $P$; if not, explain why.

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