
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
Call a set of  $n$  lines *good* if no  $3$  lines are concurrent. These  $n$  lines divide the Euclidean plane into regions (possible unbounded). A *coloring* is an assignment of two colors to each region, one from the set  $\{A_1, A_2\}$  and the other from  $\{B_1, B_2, B_3\}$ , such that no two adjacent regions (adjacent meaning sharing an edge) have the same  $A_i$  color or the same  $B_i$  color, and there is a region colored  $A_i, B_j$  for any combination of  $A_i, B_j$ .

A number  $n$  is *colourable* if there is a coloring for any set of  $n$  good lines. Find all colourable  $n$ .

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