
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
Let $m$ and $k$ be integers with $k$ odd. The function $\left[\frac{m}{k}\right]$ denotes the integer closest to $\frac{m}{k}$. For every odd integer $k$, determine the probability $P(k)$ that
\[\left[\frac{n}{k}\right] + \left[\frac{120 - n}{k}\right] = \left[\frac{120}{k}\right]\]
for an integer $n$ randomly chosen from the interval $1 \leq n \leq 120$. What is the minimum possible value of $P(k)$ over the odd integers $k$ in the interval $1 \leq k \leq 120$?
A) $\frac{1}{2}$
B) $\frac{60}{119}$
C) $\frac{49}{98}$
D) $\frac{58}{116}$
E) $\frac{M}{120}$

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