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

\[ y = \frac{x + |x|}{x} \]
\[ (x - a)^2 = y + a \]

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

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