You are a helpful assistant that can solve the given question step by step with the help of the wikipedia search tool and python interpreter tool. Given a question, you need to first think about the reasoning process in the mind and then provide the answer. During thinking, you can invoke the wikipedia search tool to search and python interpreter tool to calculate the math problem for fact information about specific topics if needed. The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags respectively, and the search query and result are enclosed within <search> </search> and <result> </result> tags respectively. For example, <think> This is the reasoning process. </think> <search> search query here </search> <result> search result here </result> <think> This is the reasoning process. </think> <python> python code here </python> <result> python interpreter result here </result> <think> This is the reasoning process. </think> <answer> The final answer is \[ oxed{answer here} \] </answer>. In the last part of the answer, the final exact answer is enclosed within oxed{} with latex format.
system
A right circular cone sits on a table, pointing upwards. The cross-section triangle, perpendicular to the base, has a vertex angle of 45 degrees. The diameter of the cone's base is $16$ inches. A cylinder is also placed co-axially inside the cone, its height equal to the height of the cone and its diameter equal to half of the cone's base diameter. A sphere is placed inside the cylinder so that it is tangent to the sides of the cylinder and to the cone's sides. What is the volume, in cubic inches, of the sphere? Express your answer in terms of $\pi$.
user
\frac{256}{3}\pi
rule
A right circular cone sits on a table, pointing upwards. The cross-section triangle, perpendicular to the base, has a vertex angle of 45 degrees. The diameter of the cone's base is $16$ inches. A cylinder is also placed co-axially inside the cone, its height equal to the height of the cone and its diameter equal to half of the cone's base diameter. A sphere is placed inside the cylinder so that it is tangent to the sides of the cylinder and to the cone's sides. What is the volume, in cubic inches, of the sphere? Express your answer in terms of $\pi$.
train