= Solution
Take
$$
\boxed{a_2=a\,b_1}.
$$
By <symbol calculus>, $a_2\in S^{-M-1}=S^{N_Q-1}$. Its zeroth-order contribution is $Qa_2=b_1$ outside the compact transition region and therefore cancels the order $-1$ remainder. Every term in
$$
Q(x,\lambda+D_x)a_2-Qa_2
$$
contains at least one $\lambda$ derivative of $Q$ and has order at most $(M-1)+(-M-1)=-2$. Absorbing these terms and the smoothing cutoff contribution into $b_2$ gives
$$
\boxed{Q(x,\lambda+D_x)(a_1+a_2)=1-b_2},\qquad b_2\in S^{-2}.
$$
Solved by gpt-5.6-sol high.
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