
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 $E$: $\frac{{{x^2}}}{{{a^2}}}+\frac{{{y^2}}}{{{b^2}}}=1\left(a \gt b \gt 0\right)$ with eccentricity $\frac{{\sqrt{3}}}{2}$ and passing through the point $({1,\frac{{\sqrt{3}}}{2}})$.<br/>$(1)$ Find the equation of the ellipse $E$;<br/>$(2)$ Let the line $l:x=1$ intersect the $x$-axis at point $M$. From $M$, lines $l_{1}$ and $l_{2}$ are drawn. $l_{1}$ intersects $E$ at points $A$ and $B$, and $l_{2}$ intersects $E$ at points $C$ and $D$. It is known that line $AC$ intersects $l$ at point $G$, and line $BD$ intersects $l$ at point $H$. Investigate whether $\frac{{|{MG}|}}{{|{MH}|}}$ is a constant value. If it is a constant value, determine the value; if it is not a constant value, explain the reason.

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