
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 \[x+y=z+u,\] \[2xy=zu.\] Find the greatest value of the real constant  $m$  such that  $m \le \frac{x}{y}$  for any positive integer solution  $(x, y, z, u)$  of the system, with  $x \ge y$ .

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