
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
For a positive integer  $k>1$  with  $\gcd(k,2020)=1,$  we say a positive integer  $N$  is *$k$ -bad* if there do not exist nonnegative integers  $x$  and  $y$  with  $N=2020x+ky$ . Suppose  $k$  is a positive integer with  $k>1$  and  $\gcd(k,2020)=1$  such that the following property holds: if  $m$  and  $n$  are positive integers with  $m+n=2019(k-1)$  and  $m \geq n$  and  $m$  is  $k$ -bad, then  $n$  is  $k$ -bad. Compute the sum of all possible values of  $k$ .

*Proposed by Jaedon Whyte*

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