
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 the parameter \( a \) for which there exists a number \( b \) such that the system

$$
\left\{\begin{array}{l}
y = x^{2} + a \\
x^{2} + y^{2} + 2 b^{2} = 2 b(x - y) + 1
\end{array}\right.
$$

has at least one solution \((x, y)\).

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