the eMSs with the modified left eigenmodes from Sec. \[ResetSpec\], the eMSs of the model with resets are given by $$\begin{aligned}
\label{MetaeMS}
\ket{\tilde{P}^\Gamma_{1}}=&\ket{{P}^\Gamma_{\rm ss}}+{c}_{2}^{\Gamma,\rm max}\ket{{r}_{2}}\nonumber\\
\ket{\tilde{P}^\Gamma_{2}}=&\ket{{P}^\Gamma_{\rm ss}}+{c}_{2}^{\Gamma,\rm min}\ket{{r}_{2}},\end{aligned}$$ where, since the modification to the left eigenmodes is simply a shift by the flat state, the coefficients are now given by $${c}_{2}^{\Gamma,\rm min/max}={c}_{2}^{\rm min/max}-\frac{\Gamma\left\langle l_2|C_0\right\rangle}{\Gamma-\lambda_2}.$$ Substituting this and Eq.  into Eq. , we see that the eMSs with reset are approximately equal to the original eMSs, as the effect of the modifications to the steady state and left eigenmodes cancel. Applying Eq.  to the model with reset and using $\ket{\tilde{P}^\Gamma_{i}}\approx\ket{\tilde{P}_{i}}$, we thus obtain $$\begin{aligned}
\label{ResetMetastableDecomp}
\ket{P^\Gamma_\text{ss}}&\approx&\left(p_1^\text{ss}+\Delta p\right)\ket{\tilde{P}_1}+\left(p_2^\text{ss}-\Delta p\right)\ket{\tilde{P}_2}\nonumber\\
&=&\ket{P_{ss}}+\Delta p \ket{\tilde{P}_1}-\Delta p\ket{\tilde{P}_2},\nonumber\end{aligned}$$ where $$\Delta p = \frac{\Gamma\left\langle l_2|C_0\right\rangle}{\Delta c_2(\Gamma-\lambda_2)}.$$ We see that, depending on the overlap of the reset state with $\ket{l_2}$, this coefficient can cause a notable modification of the steady state mixture even for small $\Gamma$. This means physically that resets will make whichever eMS is closer to the reset state more likely to occur in the stationary state, as expected. Finally, the modified effective dynamics can be constructed simply by replacing the coefficients and eigenvalues with the new ones in Eq. .

Applications {#Applications}
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We apply in 