A challenging infinite sum with limits

Let $\{ a_n \}$ satisfy the recurrence relation $a_{n+1} = (1-S_n) c^2 + c \sqrt{ (1-S_n)^2 c^2 + S_n (2-S_n) } ,$ where $a_1 = 2c^2$ , $S_n = \sum \limits_{k=1}^n a_k$ , and $0<c<1.$

Denote $X = \lim \limits_{c\to 0^+} \dfrac1c \sum \limits_{n=1}^\infty a_n ^2$ . Submit your answer as $\left \lfloor 10^{10} X \right \rfloor$ .

Notation: $\lfloor \cdot \rfloor$ denotes the floor function .