Starting with any non-zero natural number, it is always possible to form a sequence of numbers that ends in 1, by repeatedly following the instructions below:

- if the number is odd, add 1;
- if the number is even, divide by 2.

For example, starting with the number 21, the following sequence is formed:

$$
21 \rightarrow 22 \rightarrow 11 \rightarrow 12 \rightarrow 6 \rightarrow 3 \rightarrow 4 \rightarrow 2 \rightarrow 1
$$

This sequence contains nine numbers; therefore, we say it has a length of 9. Additionally, since it starts with an odd number, we say it is an odd sequence.

a) Write the sequence that starts with 37.

b) There are three sequences of length 5, two of which are even and one is odd. Write these sequences.

c) How many even sequences and how many odd sequences are there of length 6? And of length 7?

d) There are a total of 377 sequences of length 15, 233 of which are even and 144 are odd. How many sequences are there of length 16? Of these, how many are even? Do not forget to justify your answer.