
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 \(Q(k)\) be the probability that
\[
\left[\frac{n}{k}\right] + \left[\frac{150 - n}{k}\right] = \left[\frac{150}{k}\right]
\]
holds for an integer \(n\) randomly chosen from \(1 \leq n \leq 149\). Determine the minimum possible value of \(Q(k)\) over the odd integers \(k\) in the interval \(1 \leq k \leq 149\).
**A)** $\frac{22}{67}$  
**B)** $\frac{1}{3}$  
**C)** $\frac{40}{89}$  
**D)** $\frac{30}{149}$  
**E)** $\frac{1}{2}$

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