Let $f(n)$ and $g(n)$ be functions satisfying $$f(n)=\{\begin{array}{ll}\sqrt{n}& \text{ if }\sqrt{n}\text{ is an integer}\\ 1+f(n+1)& \text{ otherwise}\end{array}$$and $$g(n)=\{\begin{array}{ll}\sqrt{n}& \text{ if }\sqrt{n}\text{ is an integer}\\ 2+g(n+2)& \text{ otherwise}\end{array}$$for positive integers $n$ Find the least positive integer $n$ such that $\frac{f(n)}{g(n)}=\frac{4}{7}$.