How to to break a braced equation in the middle

Question

I'm thinking about how to break the following equation: As you can see, there is a big brace wrapping up one of the factors in the product integrand. Now I'm trying to change it into the layout in the edited picture below:

My problem is, I don't know how to properly break the factor with both braces kept in the same size. Here is my original code:

\documentclass[12pt]{amsart}
\usepackage[left=3cm,right=3cm,top=3cm,bottom=2cm]{geometry}
\usepackage{CJKutf8,mathpazo,upgreek,mathrsfs}
\usepackage{amssymb,amsthm,mathtools,mathabx,physics}
\usepackage{enumerate,nicefrac,relsize}
\usepackage{cases,csquotes}
\newcommand{\lrBra}[1]{\left\lbrace#1\right\rbrace}
\begin{document}
Suppose $\lrBra{E_t}{t>0}$ weakly solves \eqref{Eq: IMCF} with $E_0$ as the initial condition. If $E_0$ is a minimizing hull, then for $0\leq r<s$,
\begin{align}
&m(\Sigma_s)-m(\Sigma_r)\geq\frac{1}{(16\pi)^{\nicefrac{3} 
{2}}}\int_r^s|\Sigma_t|^{\nicefrac{1}{2}}\cdot\label{Eq: weak Gmf}\
&\lrBra{16\pi-8\pi\chi(\Sigma_t)+\int{\Sigma_t}\lrbra{2H^{-2}\norm{\mathrm{grad}\ 
H}^2+\mathsf{R}M+(\lambda_1-\lambda_2)^2}\dd\mu{\Sigma_t}}\dd t,\notag
\end{align}
provided that $E_s$ is precompact.

I'd like someone to help me achieve the desired layout in the edited picture. Thanks for kindly help.