= Solution
If $w=1$ in $P$, its image is the identity in the output group, so $c=1$. The equations $u_j^{-1}cu_j=g_j$ then give $z=d=1$ and $x_i z=1$ for every $i$, hence $x_i=1$. Therefore $C=1$, and the relation $C=a_1$ gives $a_1=1$. Since the <normal closure> of $a_1$ is $Q$, every $a_i$ is $1$. The remaining identifications make every $u_j$ equal to $1$ as well. All output generators vanish, proving
$$
\boxed{w=1\text{ in }P\quad\Longrightarrow\quad P(w)=\{1\}.}
$$
This proves collapse even when the original <group> $P$ itself is nontrivial.
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