Question 155
Let $n \geq 2$ be an integer. Initially, the number $1$ is written $n$ times on a board. Every minute, Vishal picks two numbers written on the board, say $a$ and $b$, erases them, and writes either $a+b$ or $\min \{a^{2}, b^{2}\}$. After $n-1$ minutes there is one number left on the board. Let the largest possible value for this final number be $f(n)$. Prove that
$$
2^{n / 3} \lt f(n) \leq 3^{n / 3}
$$
$$
2^{n / 3} \lt f(n) \leq 3^{n / 3}
$$