
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
Find the largest positive integer  $k{}$  for which there exists a convex polyhedron  $\mathcal{P}$  with 2022 edges, which satisfies the following properties:


- The degrees of the vertices of  $\mathcal{P}$  don’t differ by more than one, and
- It is possible to colour the edges of  $\mathcal{P}$  with  $k{}$  colours such that for every colour  $c{}$ , and every pair of vertices  $(v_1, v_2)$  of  $\mathcal{P}$ , there is a monochromatic path between  $v_1$  and  $v_2$  in the colour  $c{}$ .

*Viktor Simjanoski, Macedonia*

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