How many different prime numbers are factors of $N$ if

$\log_2 ( \log_3 ( \log_5 (\log_ 7 N))) = 11?$
$\mathrm{(A) \ }1 \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ }3 \qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ }7$
Re-writing as exponents, we have $\log_3 ( \log_5 (\log_ 7 N)) = 2^{11}$, and so forth, such that $N = 7^{5^{3^{2^{11}}}}$, which only has $7$ as a prime factor $\mathbf{(A)}$.