A sequence of integers$$$A $=\backslash \{$a$\_1,$ a$\_2,...,$ a$\_$n$\backslash \}$$$$is said to be$*$bumpy$*$when the signs of the differences between two consecutive terms in the sequence strictly alternate between $+$ and $-$ values.$$ A difference of zero can never be part of a$*$bumpy$*$sequence$.$ So the sequence either follows $a$\_1<$ a$\_2>$ a$\_3<$ a$\_5>...$$ $a$\_1>$ a$\_2<$ a$\_3>$ a$\_4<...$$$.$

An example of a$*$bumpy$*$sequence is $2,4,-1,9,0,5,-2.$ On the other hand, the sequence $2,4,7,3,10,5,5$ is not$*$bumpy$*$because the differences between the three consecutive elements $2,4,7$ do not alternate. Two $5\text{'}$s also show up at the end of the sequence causing the consecutive difference to be zero.

You are given a sequence of integers$$$A $=\backslash \{$a$\_1,$ a$\_2,...,$a$\_$n$\backslash \}$$$.$ Your task is to find the length of the longest$*$bumpy$*$subsequence in A$.$ Design a dynamic programming algorithm to solve this problem.