
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
Each two-digit is number is coloured in one of  $k$  colours. What is the minimum value of  $k$  such that, regardless of the colouring, there are three numbers  $a$ ,  $b$  and  $c$  with different colours with  $a$  and  $b$  having the same units digit (second digit) and  $b$  and  $c$  having the same tens digit (first digit)? 

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