Converging Triangle

Geometry Level 5

Let there be a triangle $A_1A_2A_3$ as shown in figure below. Lines $A_iA_{i+1}$ are drawn with $A_{i+1}$ lying on $A_{i-1}A_{i-2}$ such that

$\hspace{5cm}\Large\frac{A_{i-2}A_{i+1}}{A_{i+1}A_{i-1}} = \frac{3}{7}$$\hspace{1cm}\;i\geq 3\; i\in N$

In this way $\triangle A_{i-2}A_{i-1}A_i$ becomes smaller as $i$ is getting larger and ultimately converges to a point say $J$. Join $A_1$ and $J$ and extend it to meet $A_2A_3$ at $E$.

$\hspace{8cm}$ If $\Large\frac{A_1J}{JE}=\frac{a}{b}$ where $a$ and $b$ are coprime. Find $a+b$.

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