Find the value of \[\cot(\cot^{-1}3+\cot^{-1}7+\cot^{-1}13+\cot^{-1}21).\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Let's think step by step and output the final answer within \boxed{}.
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