
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
Let  $f(n)$  be the number of ones that occur in the decimal representations of all the numbers from 1 to  $n$ .  For example, this gives  $f(8)=1$ ,  $f(9)=1$ ,  $f(10)=2$ ,  $f(11)=4$ , and  $f(12)=5$ .  Determine the value of  $f(10^{100})$ .

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