use $A \lesssim_M B$ to mean $A \le CB$ where $C$ is a constant that can depend on $M$.

Preliminaries to the analysis
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Self-similar variables and derivation of equations {#sec:var}
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We will employ the notation $$\begin{aligned}
\beta_{\tau}=\frac{1}{1-\dot\tau},\quad\beta_1=\frac{1}{1+\lambda},\quad\beta_2=\frac{1-\lambda}{1+\lambda},\quad\beta_3=\frac{1-2\lambda}{1+\lambda},\quad\beta_4=\frac{3+2\lambda}{1+\lambda},\quad\beta_5=\frac{\lambda}{2+2\lambda}\,.\end{aligned}$$

We now introduce the change of coordinates that we work in and the relevant modulation variables. We define our self-similar temporal and spacial variables as $$\begin{aligned}
 \label{sec3:1}
s =- \log( \tau - t), \qquad x = \frac{\theta - \xi(t)}{(\tau - t)^{\frac 5 4}}\,.\end{aligned}$$ We record the following identities $$\begin{aligned}
&\frac{{\ensuremath{\partial}}s}{{\ensuremath{\partial}}t} = (1 - \dot{\tau}) e^s, \qquad \frac{{\ensuremath{\partial}}x}{{\ensuremath{\partial}}t} = \frac{5}{4} (1 - \dot{\tau}) x e^s - \dot{\xi} e^{\frac 5 4 s}, \qquad \frac{{\ensuremath{\partial}}x}{{\ensuremath{\partial}}\theta} = e^{\frac 5 4 s}\,.\end{aligned}$$ We now introduce the new unknowns, $W, Z, A$ which are defined through the following relations $$\begin{aligned}
 \label{sec3:2}
w(\theta, t) = e^{- \frac{s}{4}} W(x, s) + \kappa(t), \qquad z(\theta, t) = Z(x, s), \qquad a(\theta, t) = A(x, s)\,. \end{aligned}$$ In order to solve for the three modulation variables $\kappa$, $\tau$ and $\xi$, we enforce the following constraints $$\begin{aligned}
\label{e:constraints}
W(0, s) = 0, \qquad {\ensuremath{\partial}}_x W(0, s) = - 1, \qquad {\ensuremath{\partial}}_x^4 W(0, s) = 0\,.\end{aligned}$$

We now record the following calculations $$\begin{aligned}
