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
Three cones with apex $A$ touch each other externally, with the first two cones being identical and the vertex angle of the third cone equal to $2 \arcsin \frac{1}{4}$. Each of the cones touches internally a fourth cone with apex at point $A$. Find the vertex angle of the first two cones if it is half that of the fourth cone. (The vertex angle of a cone is the angle between its generatrices in an axial section.)
user
\frac{\pi}{6} + \arcsin \left(\frac{1}{4}\right)
rule
Three cones with apex $A$ touch each other externally, with the first two cones being identical and the vertex angle of the third cone equal to $2 \arcsin \frac{1}{4}$. Each of the cones touches internally a fourth cone with apex at point $A$. Find the vertex angle of the first two cones if it is half that of the fourth cone. (The vertex angle of a cone is the angle between its generatrices in an axial section.)
train