The Homecoming football game will be canceled due to rain, says the prescient Sreenath. Due to the rain, a certain tree will grow very tall. The tree is initially a single branch of length $7$. Each second, each branch less than $1$ second old will, with probability $\frac{2}{3}$, grow a random number of branches between $0$ and $7$, each with a random length from $0$ to the previous branch’s length. Find the expected height of the tree.