
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
\[
1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 \dots
\]
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 100th digit enters the sequence in the placement of the two-digit integer 55. Find, with proof,  $f(1987)$ .

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