
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
Let $\max \{a,b\}$ denote the maximum value of $a$ and $b$. When the positive numbers $x$ and $y$ (where $x > 2y$) change, find the minimum value of $t = \max\left\{\frac{{x}^{2}}{2}, \frac{4}{y(x-2y)}\right\}$.

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