
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
a) Verify that if we choose 3 or more integers from the set \(\{6k+1, 6k+2, 6k+3, 6k+4, 6k+5, 6k+6\}\), at least two will differ by 1, 4, or 5.

b) What is the largest number of positive integers less than or equal to 2022 that we can choose such that no two numbers have a difference of 1, 4, or 5?

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