
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
For a binary function $f(x, y)$, $\underset{min}{x}\{\underset{max}{y}\{f(x, y)\}\}$ represents finding the maximum value of $f(x, y)$ with respect to $y$ first, and then finding the minimum value with respect to $x$. Given non-zero vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ in the plane, satisfying: $\overrightarrow{a}⊥\overrightarrow{b}$, $\frac{\overrightarrow{a}•\overrightarrow{c}}{|\overrightarrow{a}|}=2\frac{\overrightarrow{b}•\overrightarrow{c}}{|\overrightarrow{b}|}$, let $f(m, n)=\frac{|m\overrightarrow{c}-\overrightarrow{b}|}{|m\overrightarrow{c}-n\overrightarrow{a}|}$ ($m$, $n\in \mathbb{R}$ and $m\neq 0$, $n\neq 0$), then $\underset{min}{m}\{\underset{max}{n}\{f(m, n)\}\}=$____.

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