$1.($Lexical analysis; $20$ points$)$: Java package names:

In Java source code, one can indicate the package that contains the particular class being defined, e$.$g$.$:

package edu.umass.cs$.$cs$610.$mpc;

public class mpc $\backslash \{...\backslash \}$

A package name $($e$.$g$.,$ edu.umass.cs$.$cs $\backslash (610.\backslash$mathrm$\{$mpc$\}"\backslash ))$ consists of one or more identifiers, separated by dot $(\backslash (\backslash$because $\backslash ).\backslash ()\backslash )$ characters. $($It cannot start or end in a dot, or have two dots in a row.$)$ An identifier starts with a letter $($a$-$z or A$-$Z$),$ underscore, or dollar sign, and may continue with zero or more letters, underscores, dollar signs, or digits $(0-9).$

$($a: $20$ points$)$ Draw a DFA for package names.