
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 $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with left and right foci $F_{1}$, $F_{2}$, and $P$ is a moving point on $C$ different from the left and right vertices. The minimum value of $\overrightarrow{PF_{1}} \cdot \overrightarrow{PF_{2}}$ is $2$, and the eccentricity of $C$ is $\frac{1}{2}$.  
$(1)$ Find the equation of the ellipse $C$.  
$(2)$ If a circle $E$ is tangent to all three sides of $\triangle PF_{1}F_{2}$, determine whether there exist fixed points $M$, $N$ such that $|EM|+|EN|$ is a constant value. If they exist, find the coordinates of points $M$, $N$; if not, explain the reason.

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