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The following MWE

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{align*}
        F_1(p) & = 1-\frac{L_0(\underline{c}_1-p)+\mu \underline{c}_1}{\mu p} & \text{ for } p\in \left[\frac{cL_0}{L_0+\mu},c \right] \\
        F_{02}(p) & =\begin{cases} 1-\frac{L_1\left(\frac{cL_0}{L_0+\mu}-p\right)+\mu \frac{cL_0}{L_0+\mu}}{\mu p} & \text{ for } p\in \left[\frac{cL_0}{L_0+\mu},c \right) \\ 
                                   1 & \text{ for } p\geq c,
                     \end{cases}
    \end{align*}
\end{document}

produces

enter image description here

How can I modify the code to achieve alignment at the equals signs and the "for" statements?

enter image description here

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