For a positive integer $n$, define the $n$th triangular number $T_n$ to be $\frac{n(n+1)}{2}$, and define the $n$th square number $S_n$ to be $n^2$. Find the value of 
\[ \sqrt{S_{62}+T_{63}\sqrt{S_{61}+T_{62}\sqrt{\cdots \sqrt{S_2+T_3\sqrt{S_1+T_2}}}}}. \]

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