2024 AIME I Problem 8

Eight circles of radius 34 can be placed tangent to \overline{BC} of \triangle ABC so that the circles are sequentially tangent to each other, with the first circle being tangent to \overline{AB} and the last circle being tangent to \overline{AC}, as shown. Similarly, 2024 circles of radius 1 can be placed tangent to \overline{BC} in the same manner. The inradius of \triangle ABC can be expressed as \frac{m}{n}, where m and n are relatively prime positive integers. Find m + n.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.