
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

Janete creates sequences from a 4-digit number using the following process:

I) She divides the number into two parts: the first part consists of the first two digits and the second part consists of the last two digits.

II) If the number of one of the parts is even, she divides it by 2, but if it is odd, she adds 1.

III) Then, she concatenates the two results in the original order.

IV) If the new number still has 4 digits, she repeats the process. If it has 3 digits, she separates it into two parts, the first part being the first digit and the second part being the last two digits, and repeats the process. If it has 2 digits, she repeats the process without dividing the number.

V) This process continues until she gets a number with just 1 digit, at which point the sequence ends.

For example, the sequence starting with 1,617 is:

$$
\begin{aligned}
& 1617 \rightarrow 16|17 \rightarrow 818 \rightarrow 8|18 \rightarrow 49 \rightarrow \\
& 50 \rightarrow 25 \rightarrow 26 \rightarrow 13 \rightarrow 14 \rightarrow 7
\end{aligned}
$$

Therefore, the sequence is:

$$
1617 \rightarrow 818 \rightarrow 49 \rightarrow 50 \rightarrow 25 \rightarrow 26 \rightarrow 13 \rightarrow 14 \rightarrow 7
$$

a) What is the sequence starting with 2,020?

b) What is the 5th term in the sequence starting with 8,998?

c) We define a subsequence as the last 5 numbers in a sequence. How many different subsequences exist that end in 7, but all their terms consist of 2-digit numbers, except for the final 7?

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