
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
Call a positive integer  $k$   $\textit{pretty}$  if for every positive integer  $a$ , there exists an integer  $n$  such that  $n^2+n+k$  is divisible by  $2^a$  but not  $2^{a+1}$ . Find the remainder when the  $2021$ st pretty number is divided by  $1000$ .

*Proposed by i3435*

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