
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
Suppose a convex polygon with $4n-1$ sides, where $n$ is a positive integer greater than 1. The set of all diagonals of the polygon is divided into $k$ subsets \( S_{1}, S_{2}, \cdots, S_{k} \) such that for every pair of different indices $i$ and $j$, there exist some diagonals in subset $S_{i}$ that intersect with some diagonals in subset $S_{j}$ (where intersection means intersecting at an interior point of the segments). Determine the maximum possible value of $k$.

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