Consider the following deterministic finite automaton (DFA). The number of strings of length 8 accepted by the above automaton is ________.

Let us trace the DFA from the start state $q_0$:

1. Step 1 (Length 1): On reading the first symbol ($0$ or $1$), the automaton transitions from the start state to one of the two level-1 states.

2. Step 2 (Length 2): On reading the second symbol ($0$ or $1$), transitions from any level-1 state lead to one of the two level-2 states.

3. Step 3 (Length 3): On reading the third symbol ($0$ or $1$), every transition from either level-2 state leads to the single final/accepting state (with the double circle).

4. Steps 4 to 8: The final state has a self-loop on both input symbols $\{0, 1\}$. Once reached, the automaton remains in the accepting state for any subsequent symbol.

Thus, every binary string of length $\ge 3$ is accepted by this DFA.

The total number of binary strings of length 8 is:

Since all of them are accepted, the answer is 256.