
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 and a divisor of 150, denote by $\left[\frac{m}{k}\right]$ the integer closest to $\frac{m}{k}$. For every odd integer $k$ that divides 150, let $P(k)$ be the probability that
\[\left[\frac{n}{k}\right] + \left[\frac{150 - n}{k}\right] = \left[\frac{150}{k}\right]\]
for an integer $n$ randomly chosen from the interval $1 \leq n \leq 149$. What is the minimum possible value of $P(k)$ over the odd integers $k$ out of the integer divisors of 150 in the interval $1 \leq k \leq 149$?
A) $\frac{1}{5}$
B) $\frac{1}{7}$
C) $\frac{1}{3}$
D) $\frac{1}{9}$
E) $\frac{1}{11}$

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