Let $a_1 = 11^{11}$, $a_2 = 12^{12}$, $a_3 = 13^{13}$, and 
\[a_n = |a_{n-1} - a_{n-2}| + |a_{n-2} - a_{n-3}|, \quad n \ge 4.\]
Determine $a_{14^{14}}$.

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