
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

1. Calculate the sums: \(a=99+999\), \(b=99+999+9999\) and \(c=99+999+9999+99999\).
2. Consider the number \(N\) defined as the sum:

$$
N=99+999+9999+\cdots+9999999 \ldots 999
$$

The first term in this sum is written with two digits of 9; subsequent terms are written with three, then four digits of 9, and so on. The last term in the sum is written with one hundred digits of 9. After evaluating the sum and writing \(N\) in its decimal form, how many times does the digit 1 appear in this form?

3. Answer the same question for the number \(M\) defined as:

$$
M=11+111+1111+\cdots+1111111 \ldots 111
$$

where the last term in the sum is written with one hundred digits of 1. How many times does the digit 1 appear in this form?

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