
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
The numbers \( x \) and \( y \) are solutions to the system
\[
\left\{
\begin{array}{l}
a x - y = 2a + 1 \\
-x + a y = a
\end{array}
\right.
\]
where \( a \) is a parameter. What is the minimum value of the expression \( 2y^{2} + x^{2} \) if \( a \in [-0.5, 2] \)?

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