
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 eccentricity $\frac{1}{2}$, the line $mx+y-m=0$ always passes through one focus $F$ of $E$.  
$(1)$ Find the standard equation of $E$;  
$(2)$ Let $O$ be the origin, and the vertices of quadrilateral $ABCD$ lie on $E$, $AC$ and $BD$ intersect at $F$, and $\overrightarrow{AC}⋅\overrightarrow{BD}=0$, $\overrightarrow{OA}+\overrightarrow{OC}=2\overrightarrow{OM}$, $\overrightarrow{OB}+\overrightarrow{OD}=2\overrightarrow{ON}$. If the cosine value of the slope of line $AC$ is $\frac{\sqrt{5}}{5}$, find the coordinates of the intersection point of line $MN$ and the $x$-axis.

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