Habitable Planet in the L2 Lagrange Point of a Hot Jupiter

If orbiting in the L2 lagrange point of a large planet (doesn't actually have to be a hot Jupiter, just large enough to fully shadow the smaller planet), how close could you get to the host star before the planet is no longer habitable?

If possible, I'd like to ignore any stability issues. The L2 lagrange point of Earth is far too unstable for a natural body; the satellites we put there must regularly make slight adjustments to stay in place. I'm not sure if substantially increasing the size of the large planet would help with stability (or even make it worse) or if there is some way to mitigate this, but that is another issue.

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Habitable Planet in the L2 Lagrange Point of a Hot Jupiter

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