You are a helpful assistant. When the user asks a question, your response must include two parts: first, the reasoning process enclosed in <think>...</think> tags, then the final answer enclosed in <answer>...</answer> tags.Please provide a clear, concise response within <answer> </answer> tags that directly addresses the question.
system
<image>There are exactly three positive real numbers $k$ such that the function
$$f(x) = \frac{(x - 18)(x - 72)(x - 98)(x - k)}{x}$$
defined over the positive real numbers achieves its minimum value at exactly two positive real numbers $x$. Find the sum of these three values of $k$. You FIRST think about the reasoning process as an internal monologue and then provide the final answer. The reasoning process MUST BE enclosed within <think> </think> tags. Please answer the final answer within the <answer> </answer> tags.
user
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There are exactly three positive real numbers $k$ such that the function
$$f(x) = \frac{(x - 18)(x - 72)(x - 98)(x - k)}{x}$$
defined over the positive real numbers achieves its minimum value at exactly two positive real numbers $x$. Find the sum of these three values of $k$. You FIRST think about the reasoning process as an internal monologue and then provide the final answer. The reasoning process MUST BE enclosed within <think> </think> tags. Please answer the final answer within the <answer> </answer> tags.
val