
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<sub>1</sub>: 2x - y + a = 0 (a > 0), l<sub>2</sub>: -4x + 2y + 1 = 0, and l<sub>3</sub>: x + y - 1 = 0, and the distance between l<sub>1</sub> and l<sub>2</sub> is $\frac{7\sqrt{5}}{10}$;
(1) Find the value of a;
(2) Is there a point P that simultaneously satisfies the following three conditions:
① P is a point in the first quadrant;
② The distance from point P to l<sub>1</sub> is half of the distance from P to l<sub>2</sub>;
③ The ratio of the distance from point P to l<sub>1</sub> to the distance from P to l<sub>3</sub> is $\sqrt{2}$: $\sqrt{5}$. If so, find the coordinates of point P; if not, explain why.

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