
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
The sequence of digits \[123456789101112131415161718192021\ldots\] is obtained by writing the positive integers in order. If the  $10^n$ th digit in this sequence occurs in the part of the sequence in which the  $m$ -digit numbers are placed, define  $f(n)$  to be  $m$ . For example,  $f(2) = 2$  because the  $100^{\text{th}}$  digit enters the sequence in the placement of the two-digit integer  $55$ . Find the value of  $f(2007)$ .

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