
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
A hunter first shoots at a fox running away from him from a distance of $100 \, m$; let the probability of hitting the fox from this distance be $\frac{1}{2}$ (i.e., from a distance of $100 \, m$, the hunter is equally likely to hit or miss the running fox). In case of a miss, the hunter reloads the gun and shoots again, but during this time, the fox manages to move away an additional $50 \, m$. In case of a second miss, he reloads the gun again and shoots a third (and last) time, during which the fox will move another $50 \, m$ away. Assuming the probability of hitting the fox is inversely proportional to the square of the distance, determine the probability that the hunter will be able to hit the fox.

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