
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 the plane of Camelot, King Arthur built a labyrinth $\mathscr{L}$ consisting of $n$ walls. Each wall is a straight line (infinitely extended). No two walls are parallel, and no three walls have a common point. The wizard Merlin painted one side of each wall red and the other side blue.

At the intersection of two walls, there are four angles: two opposite angles are formed by one red side and one blue side; one angle is formed by two red sides; one angle is formed by two blue sides. At each such intersection, there is a bidirectional door leading to the two opposite angles formed by edges of different colors.

Once Merlin finished painting each wall, the sorceress Morgana places some knights in the labyrinth $\mathscr{B}$. Knights can pass through doors but cannot cross walls.

Let $k(\mathscr{D})$ be the maximum positive integer $k$ such that regardless of how Merlin colors the walls of the labyrinth $\mathscr{L}$, Morgana can arrange at least $k$ knights so that no two knights ever meet. For each positive integer $n$, find all possible values of $k(\mathscr{D})$ where the labyrinth $\mathscr{E}$ has $n$ walls.

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