
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 the parabolas ${Γ}_{1}:{y}^{2}=4x$, ${Γ}_{2}:{y}^{2}=2x$, and the line $l$ intersects the parabola $\Gamma _{1}$ at points $A$ and $D$, and intersects the parabola $\Gamma _{2}$ at points $B$ and $C$, where points $A$ and $B$ are in the first quadrant.<br/>$(1)$ If the distance from point $A$ to the focus of the parabola $\Gamma _{1}$ is $2$, find the coordinates of point $A$;<br/>$(2)$ If the coordinates of point $A$ are $\left(4,4\right)$, and the midpoint of segment $AC$ lies on the $x$-axis, find the distance from the origin $O$ to the line $l$;<br/>$(3)$ If $\overrightarrow{AB}=2\overrightarrow{CD}$, find the ratio of the areas of $\triangle AOD$ and $\triangle BOC$.

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