Write a Python program that can take different numbers of strings (each string may include digits, characters, and special symbols) and then sorted them. You need first define the rule for sorting and then implement the sorting function using Python (DO NOT use any existing sort function). Visualize a list of input strings (the list must include at least 500 strings) using a scatter plot (one dimension will be the length of each string, and the other dimension could be the order of the string in the list).