
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
Ellipse $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ divides the circumference of the circle ${x}^{2}+{y}^{2}=\frac{8}{5}$ into four equal parts, and the eccentricity of ellipse $C$ is $\frac{\sqrt{3}}{2}$.  
$(1)$ Find the equation of ellipse $C$;  
$(2)$ If the line $l$ intersects ellipse $C$ at two distinct points $M$ and $N$, and the midpoint of $MN$ is $P({x}_{0},\frac{1}{4})$, and the perpendicular bisector of segment $MN$ is ${l'}$, and line ${l'}$ intersects the $x$-axis at point $Q\left(m,0\right)$, find the range of values for $m$.

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