Simple periods

The period of an LFSR can depend on the choice of the initialization vector. Determine all possible periods of LFSRs defined by the equation:

1. $s_i = (s_{i-1} + s_{i-2} + s_{i-3}) \bmod{2}$ for all initialization vectors $(s_0, s_1, s_2) \in \Z_2^3$, and
2. $s_i = (s_{i-1} + s_{i-2} + s_{i-3} + s_{i-4}) \bmod{2}$ for all initialization vectors $(s_0, s_1, s_2, s_3) \in \Z_2^4$.

Answer the following questions with a number:

• How many initialization vectors for the first LFSR produce a sequence with period 1?
• How many initialization vectors for the first LFSR produce a sequence with period 4?
• How many initialization vectors for the second LFSR produce a sequence with period 1?
• What is the maximum possible period that the second LFSR can produce?
• Can you find an LFSR over $\Z_2$ of degree 3 that has period 7? Answer with yes or no.

Submit the answer as a sequence like 1 2 3 yes.