
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 integer values of the parameter \(a\) for which the system has at least one solution:

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

In the answer, indicate the sum of the found values of parameter \(a\).

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