
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 $min|a, b|$ denote the minimum value between $a$ and $b$. When positive numbers $x$ and $y$ vary, let $t = min|2x+y, \frac{2y}{x^2+2y^2}|$, then the maximum value of $t$ is ______.

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