
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 three lines $l_{1}:2x-y+a=0$, line $l_{2}:-4x+2y+1=0$ and line $l_{3}:x+y-1=0$, and the distance between $l_{1}$ and $l_{2}$ is $\frac{7}{10}\sqrt{5}$.

$(1)$ Find the value of $a$;

$(2)$ Is it possible to find a point $P$ that simultaneously satisfies the following three conditions:
$(i)$ $P$ is a point in the first quadrant;
$(ii)$ The distance from point $P$ to $l_{1}$ is half the distance from $P$ to $l_{2}$;
$(iii)$ The ratio of the distance from point $P$ to $l_{1}$ to the distance from $P$ to $l_{3}$ is $\sqrt{2}:\sqrt{5}$? If possible, find the coordinates of point $P$; if not, explain why.

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