
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
Consider the system
$$
\begin{aligned}
x+y & =z+u, \\
2xy & =zu.
\end{aligned}
$$

Find the greatest value of the real constant \( m \) such that \( m \leq x / y \) for every positive integer solution \( x, y, z, u \) of the system with \( x \geq y \).

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