
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  $k$  for which there exists a permutation  $(a_1, a_2, \ldots, a_{2022})$  of integers from  $1$  to  $2022$  such that for at least  $k$  distinct  $i$  with  $1 \le i \le 2022$  the number  $\frac{a_1 + a_2 + \ldots + a_i}{1 + 2 + \ldots + i}$  is an integer larger than  $1$ .

*(Proposed by Oleksii Masalitin)*

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