For positive integers $a$ and $b$, the operation $a \uparrow\uparrow b$ is defined as follows:  
- $a \uparrow\uparrow 1 = a$  
- $a \uparrow\uparrow b = a^{a \uparrow\uparrow (b-1)}$ for $b > 1$.  

Find the smallest positive integer $n$ for which there exists a positive integer $a$ such that $a \uparrow\uparrow 6 \not\equiv a \uparrow\uparrow 7 \pmod{n}$.

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