
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
Determine the number of permutations  $a_1, a_2, \dots, a_n$  of  $1, 2, \dots, n$  such that for every positive integer  $k$  with  $1 \le k \le n$ , there exists an integer  $r$  with  $0 \le r \le n - k$  which satisfies
\[ 1 + 2 + \dots + k = a_{r+1} + a_{r+2} + \dots + a_{r+k}. \]

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