
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 m and k integers with k odd, denote by [m/k] the integer closest to m/k. For every odd integer k, let P(k) be the probability that [n/k] + [100-n/k] = [100/k] for an integer n randomly chosen from the interval 1 ≤ n ≤ 99. What is the minimum possible value of P(k) over the odd integers k in the interval 1 ≤ k ≤ 99?

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