
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
Given the system of equations:
$$
\begin{cases}
x - 2y = z - 2u \\
2yz = ux
\end{cases}
$$
for each set of positive real number solutions \{x, y, z, u\}, where $z \geq y$, there exists a positive real number $M$ such that $M \leq \frac{z}{y}$. Find the maximum value of $M$.

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