
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
On a flat plane, there is a labyrinth $\mathfrak{L}$ consisting of $n$ walls, each of which is an infinite straight line. No two walls are parallel, and no three walls intersect at a common point. One side of each wall is painted red, and the other side is painted blue. At the intersection of two walls, there are four corners: two diagonally opposite corners where a red side and a blue side meet, one corner where two red sides meet, and one corner where two blue sides meet. A two-way door connects the two diagonally opposite corners where sides of different colors meet. Knights can walk through doors but cannot walk through walls. 

Let $k(\mathfrak{L})$ be the largest number $k$ such that, no matter how the labyrinth is painted, at least $k$ knights can always be placed such that no two knights can ever meet. For each $n$, what are all possible values for $k(\mathfrak{L})$, where $\mathfrak{L}$ is a labyrinth with $n$ walls?

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