Polygon Interior Angle Calculation

Consider a convex polygon with n sides, where the lengths of the sides are given by the sequence: 3, 4, 5, ..., n + 2. The polygon is arbitrary but is defined to have integer side lengths. Determine how many distinct values of n result in the polygon having an interior angle greater than $120^{\circ}$.

Recall that the interior angles of a polygon can be calculated using the formula:

$$ \text{Interior Angle} = \f\frac{(n-2) \times 180}{n} $$

Furthermore, an angle is considered greater than $120^{\circ}$ if it satisfies:

$$ \f\frac{(n-2) \times 180}{n} > 120 $$