The Power of All Powers

\[\large{\begin{cases} 5^a &\equiv& b \pmod 7 \\ 3^b &\equiv& c \pmod 5 \\ 2^c &\equiv& d \pmod 3 \\ 7^{abcd} &\equiv& 7 \pmod{11} \end{cases} } \]

Given that \(a,b,c,d\) are positive integers larger than 1 that satisfy all the congruences above, what is the least possible value of \(a\times b\times c\times d\)?

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