Santiago, Daniel and Fátima practice for the Math Olympics. Santiago thinks of a regular polygon and Daniel of another, without telling Fatima what the polygons are. They just tell you that one of the polygons has $3$ more sides than the other and that an angle of one of the polygons measures $10$ degrees more than one angle of the other. From this, and knowing that each interior angle of a regular polygon of $n$ sides measures $\frac{180(n-2)}{n}$ degrees, Fatima identifies what the polygons are. How many sides do the polygons that James and Daniel chose, have?