
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
In a flat plain at Camelot, King Arthur builds a labyrinth $\mathfrak{L}$ consisting of $n$ lines such that no two lines are parallel and no three lines are concurrent. For each wall, Merlin colors one side red and the other side blue.

At the intersection of two walls, there are 4 corners: two corners where a red side and a blue side intersect, one corner where two red sides intersect, and one corner where two blue sides intersect. At this intersection, there exists a door connecting the two opposite corners where sides of different colors intersect.

After Merlin paints the walls, Morgana places knights in the labyrinth. The knights can walk through the doors but cannot pass through walls.

Let $k(\mathfrak{L})$ be the largest integer $k$ such that, no matter how Merlin colors the labyrinth $\mathfrak{L}$, Morgana can always place $k$ knights in such a way that no two of them can ever meet. For a fixed $n$, what are the possible values of $k(\mathfrak{L})$ for labyrinths $\mathfrak{L}$ consisting of $n$ lines?

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