
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 that the point $({1,\frac{3}{2}})$ lies on the ellipse $E$: $\frac{{{x^2}}}{{{a^2}}}+\frac{{{y^2}}}{{{b^2}}}=1(a＞b＞0)$, and the eccentricity of $E$ is $\frac{1}{2}$.  
$(1)$ Find the equation of $E$;  
$(2)$ Let $F$ be the right focus of the ellipse $E$, and point $P\left(m,n\right)$ is any point on $E$. The line $PF$ intersects the line $3mx+4ny=0$ at point $Q$. Find the value of $|PQ|$.

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