
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
Find all values of \( a \) for which the system of equations

$$
y-2=a(x-4), \quad \frac{2 x}{|y|+y}=\sqrt{x}
$$

has at least one solution, and solve it for each \( a \).

Domain: \( y > 0 \), \( x \geq 0 \).

In the domain, the second equation becomes: \( x = y \sqrt{x} \).
I. \( x = 0, y = 2 - 4a > 0 \), hence \( a < \frac{1}{2} \).

II. \( x > 0, y = \sqrt{x} \); \(\sqrt{x} - 2 = a(\sqrt{x} - 2)(\sqrt{x} + 2)\).

1) \( \sqrt{x} = 2, x = 4, y = 2, a \in \mathbb{R} \).
2) \( 1 = a(\sqrt{x} + 2), \sqrt{x} = \frac{1}{a} - 2 = \frac{1 - 2a}{a} > 0, 0 < a < \frac{1}{2} \).

The found solution \( x = \left( \frac{1 - 2a}{a} \right)^2, y = \frac{1 - 2a}{a} \) coincides with the previous one if \( 2 = \frac{1}{a} - 2, a = \frac{1}{4} \).

Thus, for \( a \in \left( 0 ; \frac{1}{4} \right) \cup \left( \frac{1}{4} ; \frac{1}{2} \right) \), \( x = \left( \frac{1 - 2a}{a} \right)^2, y = \frac{1 - 2a}{a} \).

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