
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 function \( f(x) \):
\[ f(x) = \begin{cases} 
\ln x & \text{if } x > 1, \\ 
\frac{1}{2} x + \frac{1}{2} & \text{if } x \leq 1 
\end{cases} \]
If \( m < n \) and \( f(m) = f(n) \), what is the minimum value of \( n - m \)?

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