
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 all natural numbers   $k$  such that there exist natural numbers  $a_1,a_2,...,a_{k+1}$  with  $ a_1!+a_2!+... +a_{k+1}!=k!$ Note that we do not consider  $0$  to be a natural number.

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