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
Equilateral triangle $\vartriangle ABC$ has side length $12$ and equilateral triangles of side lengths $a, b, c < 6$ are each cut from a vertex of $\vartriangle ABC$, leaving behind an equiangular hexagon $A_1A_2B_1B_2C_1C_2$, where $A_1$ lies on $AC$, $A_2$ lies on $AB$, and the rest of the vertices are similarly defined. Let $A_3$ be the midpoint of $A_1A_2$ and define $B_3$, $C_3$ similarly. Let the center of $\vartriangle ABC$ be $O$. Note that $OA_3$, $OB_3$, $OC_3$ split the hexagon into three pentagons. If the sum of the areas of the equilateral triangles cut out is $18\sqrt3$ and the ratio of the areas of the pentagons is $5 : 6 : 7$, what is the value of $abc$? The original answer is in the form $k\sqrt{m}$, please give the value of k + m.
user
67
rule
Equilateral triangle $\vartriangle ABC$ has side length $12$ and equilateral triangles of side lengths $a, b, c < 6$ are each cut from a vertex of $\vartriangle ABC$, leaving behind an equiangular hexagon $A_1A_2B_1B_2C_1C_2$, where $A_1$ lies on $AC$, $A_2$ lies on $AB$, and the rest of the vertices are similarly defined. Let $A_3$ be the midpoint of $A_1A_2$ and define $B_3$, $C_3$ similarly. Let the center of $\vartriangle ABC$ be $O$. Note that $OA_3$, $OB_3$, $OC_3$ split the hexagon into three pentagons. If the sum of the areas of the equilateral triangles cut out is $18\sqrt3$ and the ratio of the areas of the pentagons is $5 : 6 : 7$, what is the value of $abc$? The original answer is in the form $k\sqrt{m}$, please give the value of k + m.
train