Let $\mathrm{A}_1, \mathrm{~A}_2, \mathrm{~A}_3$, $\_\_\_\_$ be squares such that for each $n \geq 1$, the length of the side of $A_n$ equals the length of diagonal of $A_{n+1}$. If the length of $A_1$ is 12 cm , then the smallest value of $n$ for which area of $A_n$ is less than one, is $\_\_\_\_$ .