A sequence $\{a_n\}$ with a natural number as its first term satisfies
$$a_{n+1} = \begin{cases} 2^{a_n} & (\text{if } a_n \text{ is odd}) \\ \frac{1}{2}a_n & (\text{if } a_n \text{ is even}) \end{cases}$$
for all natural numbers $n$. Find the sum of all values of $a_1$ such that $a_6 + a_7 = 3$. [4 points]\\
(1) 139\\
(2) 146\\
(3) 153\\
(4) 160\\
(5) 167