
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 $l_1: 2x - y + a = 0$ $(a > 0)$, $l_2: -4x + 2y + 1 = 0$, and $l_3: x + y - 1 = 0$, and the distance between $l_1$ and $l_2$ is $\frac{7}{10}\sqrt{5}$.
1. Determine the value of $a$;
2. Calculate the angle $\theta$ between $l_3$ and $l_1$;
3. Determine whether there is a point $P$ that simultaneously satisfies the following three conditions: ①$P$ is in the first quadrant; ②The distance from $P$ to $l_1$ is half the distance from $P$ to $l_2$; ③The ratio of the distance from $P$ to $l_1$ to the distance from $P$ to $l_3$ is $\sqrt{2} : \sqrt{5}$. If such a point exists, find the coordinates of $P$; if not, explain why.

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