
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
From the set $\{1, 2, 3, \ldots, 20\}$, choose $k$ distinct numbers to form a set $M$, such that there are always 4 distinct numbers $a, b, c, d$ in $M$ which satisfy $a - b + c - d$ being divisible by 20. Find the smallest value of $k$.

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