
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
Express \( z \) from the second equation and substitute it into the first equation:

$$
\begin{aligned}
& z = a\left(x + 2y + \frac{5}{2}\right) \rightarrow x^{2} + y^{2} + 2x - y + a\left(x + 2y + \frac{5}{2}\right) = 0 \rightarrow \\
& \rightarrow\left(x + \frac{a + 2}{2}\right)^{2} + \left(y + \frac{2a - 1}{2}\right)^{2} = \frac{(a + 2)^{2}}{4} + \frac{(2a - 1)^{2}}{4} - \frac{5a}{2}
\end{aligned}
$$

The unique solution can exist if \(\frac{(a + 2)^{2}}{4} + \frac{(2a - 1)^{2}}{4} - \frac{5a}{2} = 0 \rightarrow (a - 1)^{2} = 0 \rightarrow a = 1\).

For the parameter \( a = 1 \), the solution corresponds to \( x = -\frac{a + 2}{2} = -\frac{3}{2}, y = -\frac{2a - 1}{2} = -\frac{1}{2}, z = x + 2y + \frac{5}{2} = 0 \).

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