
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

For which values of the parameter \( a \) are the following two systems of equations equivalent:

System I:
\[ 
\begin{align*}
x^2 + y^2 &= a \\
x + y &= 0
\end{align*}
\]

System II:
\[ 
\begin{align*}
x^2 + y^2 &= a \\
\{x + y\} &= 0
\end{align*}
\]

where \( \{z\} = z - \lfloor z \rfloor \), and \( \lfloor z \rfloor \) is the greatest integer less than or equal to \( z \).

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