
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 sequence of positive integers \( a_1, a_2, a_3, \ldots \) satisfies
\[ a_{n+1} = n \left\lfloor \frac{a_n}{n} \right\rfloor + 1 \]
for all positive integers \( n \). If \( a_{30} = 30 \), how many possible values can \( a_1 \) take? (For a real number \( x \), \( \lfloor x \rfloor \) denotes the largest integer that is not greater than \( x \).)

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