
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 $k$, let $P(k)$ be the probability that
\[
\left[\frac{n}{k}\right] + \left[\frac{200 - n}{k}\right] = \left[\frac{200}{k}\right]
\]
for an integer $n$ randomly chosen from the interval $1 \leq n \leq 199$. Determine the minimum possible value of $P(k)$ over the odd integers $k$ in the interval $1 \leq k \leq 199$.
A) $\frac{1}{2}$
B) $\frac{50}{199}$
C) $\frac{44}{87}$
D) $\frac{34}{67}$ 
E) $\frac{7}{13}$

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