
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 function $f(x)$, if there exists a function $g(x)=ax+b$ (where $a$ and $b$ are constants) such that for all real numbers $x$ in the interval $D$, the inequality $f(x) \leq g(x)$ holds, then the function $g(x)$ is called a "covering function" for $f(x)$ on interval $D$. Let $f(x) = 2^x$ and $g(x) = 2x$. If the function $g(x)$ is a "covering function" for $f(x)$ on the interval $[m, n]$, then the maximum value of $2^{|m-n|}$ is $\boxed{2}$.

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