It turns out that many songs share similar chord sequences. Jazz songs in C major, for example, often

contain a sequence of chords $C^6,A^{-7},D^{-7},G^7,C^6.($By the way, even though music theory is cool,

you definitely do not need to know what these symbols mean to solve this problem!$)$

Suppose you have a database consisting of the full sequence of chords in each of a large number of

songs, and you want to learn about the common chord sequences that appear in these songs to help

you practice. One operation you might want to build into your song$-$analysis software is this: given the

full chord sequence for each of two songs, find the longest shared contiguous subsequence of chords.

To simplify matters, let's imagine we have given each possible chord a one$-$letter name. Starting with

the chords listed above, we might say $A=C^6,B=A^{-7},$ etc., so that the sequence above would be rep$-$

resented by $A,B,C,D,A.$ Further, let's just try to find the length of the longest shared contiguous

subsequence rather than the subsequence itself. For example, if your two songs are $A,B,C,D,A,D,A$

and $E,F,G,F,C,D,A,F,D,A,G,$ the length of the longest shared contiguous subsequence is $3,$ for

$C,D,A$

Write a dynamic programming algorithm to compute, given two sequences of chords $($one of length $m$

and the other of length $n),$ the length of the longest shared contiguous subsequence. As usual, provide

an argument for your algorithm's correctness and analyze its runtime.