Two Looming Towers Of Powers

Let \( a_{n+1} = 2^{a_n} \) and \( b_{n+1} = 3^{b_n} \) both for \( n \ge 1.\)

If \( a_1 = 2 \) and \(b_1 = 3\), then find \[ \min_{x\ge 2, y \ge 2} |a_x-b_y|. \]

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