Suppose real numbers $a, b, c, d$ satisfy the following conditions:

1. $a + b + c + d = 17$
2. $ab + bc + cd + da = 46$

Determine the minimum possible value of $a^2 + b^2 + c^2 + d^2$. If this value can be expressed as a rational number $\frac{p}{q}$ in its simplest form, find the sum $p + q$.

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