
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
It is known that \( m, n, \) and \( k \) are distinct natural numbers greater than 1, the number \( \log_{m} n \) is rational, and additionally,

$$
k^{\sqrt{\log_{m} n}} = m^{\sqrt{\log_{n} k}}
$$

Find the minimum possible value of the sum \( k + 5m + n \).

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