
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 $m$, $n$, and $t$ be positive real numbers satisfying the equation $m^{2} - 3mn + 4n^{2} - t = 0$. When $\frac{t}{mn}$ reaches its minimum value, find the maximum value of $m + 2n - t$.

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