
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
Find the largest  $K$  satisfying the following:

Given any closed intervals  $A_1,\ldots, A_N$  of length  $1$  where  $N$  is an arbitrary positive integer. If their union is  $[0,2021]$ , then we can always find  $K$  intervals from  $A_1,\ldots, A_N$  such that the intersection of any two of them is empty.

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