
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 the ellipse $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1(a > b > 0)$, the line $x= \dfrac {a^{2}}{c}$ (where $c$ is half the focal length of the ellipse) intersects the $x$-axis at point $A$, the top vertex of the ellipse is $B$, a line perpendicular to the $x$-axis passing through the right focus $F$ of the ellipse intersects the ellipse in the first quadrant at point $P$, and intersects $AB$ at point $D$. If point $D$ satisfies $2 \overrightarrow{OD}= \overrightarrow{OF}+ \overrightarrow{OP}$ (where $O$ is the origin).
- $(I)$ Find the eccentricity of the ellipse;
- $(II)$ If the semi-focal distance is $3$, and a line $l$ passing through point $A$ intersects the ellipse at two points $M$ and $N$, is there a fixed point $C$ on the $x$-axis such that $\overrightarrow{CM} \cdot \overrightarrow{CN}$ is constant? If it exists, find the coordinates of point $C$ and the value of the constant; if not, explain why.

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