
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 a major axis length of $4$. Point $P$ moves on $C$, $F_{1}$ and $F_{2}$ are the two foci of $C$, and $\cos \angle F_{1}PF_{2}$ has a minimum value of $\frac{1}{2}$.  
$(1)$ Find the equation of $C$;  
$(2)$ Given a moving line $l$ passing through point $M(0,m)(-b < m < b)$, intersecting $C$ at points $A$ and $B$, with the midpoint of segment $AB$ being $N$, if $\overrightarrow{OA} \cdot \overrightarrow{OB} - \overrightarrow{OM} \cdot \overrightarrow{ON}$ is a constant value, find the values of $m$.

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