
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 $E:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with left and right foci $F_{1}$, $F_{2}$, eccentricity $\frac{1}{2}$, and the maximum distance from a point on the ellipse $E$ to the focus is $3$. <br/>$(1)$ Find the equation of the ellipse $E$. <br/>$(2)$ Let $A$, $B$ be two distinct points symmetric about the $x$-axis on the ellipse $E$, $P$ be a point on the ellipse $E$ different from points $A$ and $B$, $O$ be the origin, line $AP$ intersects the $x$-axis at point $M$, line $BP$ intersects the $x$-axis at point $N$, is $|OM|\cdot |ON|$ a constant value? If it is a constant value, find this constant value; if it is not a constant value, please explain the reason.

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