Find the number of solutions to
\[\sin x = \left( \frac{1}{2} \right)^x\]on the interval $(0,100 \pi).$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Find the number of solutions to
\[\sin x = \left( \frac{1}{2} \right)^x\]on the interval $(0,100 \pi).$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
100
Find the number of solutions to
\[\sin x = \left( \frac{1}{2} \right)^x\]on the interval $(0,100 \pi).$
train