
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 square wall has side length \( n \) metres. Guillaume paints \( n \) non-overlapping circular targets on the wall, each with radius 1 metre. Mathilde is going to throw a dart at the wall. Her aim is good enough to hit the wall at a single point, but poor enough that the dart will hit a random point on the wall. What is the largest possible value of \( n \) so that the probability that Mathilde's dart hits a target is at least \( \frac{1}{2} \) ?

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