
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
Consider a $7$-digit positive integer which is considered jet-set if its $7$ digits are an arrangement of the set $\{1,2,...,7\}$ such that the integer formed by its first $k$ digits is divisible by $k$ for $k = 1,2,...,7$. Find the number of jet-set $7$-digit integers.

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