Step 1: Write OUT as a product of two blocks.
From the gate diagram, $OUT = N\cdot M$, where $N=\overline{x+y+z}$ is the 3-input NOR of $x,y,z$, and $M$ is an OR of two AND terms built from $x,y,z$ and one inverted signal, say $M=\bar z\,y + xy$.
Step 2: Build the truth table for $N$.
$N$ is $1$ only for the single row $x=0,y=0,z=0$; for the other seven rows $N=0$.
Step 3: Fill in $OUT$ for the seven rows where $N=0$.
For any row other than $(0,0,0)$, $OUT=N\cdot M=0\cdot M=0$ regardless of $M$'s value, so all seven of these rows give $OUT=0$.
Step 4: Fill in the last row, $(x,y,z)=(0,0,0)$.
Here $N=1$. Compute $M=\bar z\,y+xy=(1)(0)+(0)(0)=0$. So $OUT=1\cdot0=0$ for this row too.
Step 5: Read off the full truth table.
$$OUT = 0 \text{ for all 8 rows of } x,y,z.$$
Step 6: Interpret this result.
A function that is $0$ for every possible input is the constant function $0$, not a function like $\bar x\,\overline{yz}$ or $\bar x(y+z)$ or $1$, since each of those would be $1$ for at least one input row once the NOR gate forces that row's $N$ to the correct value.
\[ \boxed{OUT=0} \]
