
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 
\[ y - 1 = a(x - 1) \]
\[ \frac{2x}{|y| + y} = \sqrt{x} \]
has at least one solution, and solve the system for each value of \( a \).

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