Consider the following augmented grammar with $\{\#, @, \lt, \gt, a, b, c\}$ as the set of terminals. $$S^{\prime} \to S$$ $$S \to S\#cS$$ $$S \to SS$$ $$S \to S@$$ $$S \to \lt S \gt$$ $$S \to a$$ $$S \to b$$ $$S \to c$$ Let $I_0 = \text{CLOSURE}(\{S^{\prime} \to \bullet S\})$. The number of items in the set $\text{GOTO}(\text{GOTO}(I_0, \lt), \lt)$ is __________.

1. Find set $I_0 = \text{CLOSURE}(\{S^{\prime} \to \bullet S\})$:

Since the dot is before non-terminal $S$, we add all $S$-productions with the dot at the beginning:

• $S^{\prime} \to \bullet S$

• $S \to \bullet S\#cS$

• $S \to \bullet SS$

• $S \to \bullet S@$

• $S \to \bullet \lt S \gt$

• $S \to \bullet a$

• $S \to \bullet b$

• $S \to \bullet c$

2. Find set $I_1 = \text{GOTO}(I_0, \lt)$:

In $I_0$, the only item with the dot immediately preceding $\lt$ is $S \to \bullet \lt S \gt$. Advancing the dot gives the kernel item:

Taking the closure of this kernel item requires adding all productions of $S$ with the dot at the beginning:

• $S \to \lt \bullet S \gt$ (kernel item)

• $S \to \bullet S\#cS$

• $S \to \bullet SS$

• $S \to \bullet S@$

• $S \to \bullet \lt S \gt$

• $S \to \bullet a$

• $S \to \bullet b$

• $S \to \bullet c$

3. Find set $I_2 = \text{GOTO}(I_1, \lt)$:

In $I_1$, the only item with the dot immediately preceding $\lt$ is again $S \to \bullet \lt S \gt$. Advancing the dot yields the exact same kernel item:

Taking the closure results in the same 8 items as in $I_1$:

1. $S \to \lt \bullet S \gt$

2. $S \to \bullet S\#cS$

3. $S \to \bullet SS$

4. $S \to \bullet S@$

5. $S \to \bullet \lt S \gt$

6. $S \to \bullet a$

7. $S \to \bullet b$

8. $S \to \bullet c$

Thus, the total number of items in $\text{GOTO}(\text{GOTO}(I_0, \lt), \lt)$ is 8.