
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
There are 20 telephones connected with wires, with each wire connecting two telephones. Each pair of telephones is connected by at most one wire, and each telephone is connected by at most 3 wires. To color these wires so that no two wires connected to the same telephone share the same color, what is the minimum number of different colors needed?

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