
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
Given an integer  $n$ , an integer  $1 \le a \le n$  is called  $n$ -*well* if \[ \left\lfloor\frac{n}{\left\lfloor n/a \right\rfloor}\right\rfloor = a. \] Let  $f(n)$  be the number of  $n$ -well numbers, for each integer  $n \ge 1$ . Compute  $f(1) + f(2) + \ldots + f(9999)$ .

*Proposed by Ashwin Sah*

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