Find an upper bound for $T(n)=T(\sqrt{n})+10\log\log n$

Question

I need to find an upper bound for $T(n)=T(\sqrt{n})+10\log\log n$.

I thought first to make a substitution: $m=\log n$. Then: $$ T(2^m)=T(2^{m \over 2})+10\log m $$ Let $S(m)=T(2^m)$: $$ S(m)=S\big({m \over 2}\big)+10\log m $$

Now we can use Master theorem.

Suppose it's the case when $f(m)=O(m^{\log_2 1-\epsilon})=O(m^{0-\epsilon})$. Let $\epsilon=0.5$ then: $$ \lim_{n\to \infty}\frac{10\log m}{m^{0.5}}=0 $$ Hence according to the Master theorem $T(m)=\Theta(m^0)=\Theta(1)$.

Am I in the right direction?