
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
Given the functions $f\left( x \right)={e}^{4x-1},g\left( x \right)=\frac{1}{2}+{\ln }\left( 2x \right)$, if $f\left( m \right)=g\left( n \right)$ holds true, then the minimum value of $n-m$ is (   )

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