
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 every odd integer \( k \) within the range \( 1 \leq k \leq 49 \), let \( Q(k) \) represent the probability that
\[
\left[\frac{n}{k}\right] + \left[\frac{80 - n}{k}\right] = \left[\frac{80}{k}\right]
\]
holds true for an integer \( n \) randomly selected from the set \( \{1, 2, ..., 50\} \). Determine the minimum possible value of \( Q(k) \) over the specified range of \( k \).

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