
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
In an exhibition there are  $100$  paintings each of which is made with exactly  $k$  colors. Find the minimum possible value of  $k$  if any  $20$  paintings have a common color but there is no color that is used in all paintings.

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