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Suppose $\kappa$ is an inaccessible cardinal, and let $P$ be the $< \aleph_1-$support product of $Add(\alpha^{++}, 1)$ for singular cardinals $\alpha < \kappa.$ 1- Does this forcing preserve cardinals?

2-(A weaker question) Does $\kappa$ remain inaccessible in the generic extension?

Suppose $\kappa$ is an inaccessible cardinal, and let $P$ be the $< \aleph_1-$support product of $Add(\alpha^{++}, 1)$ for singular cardinals $\alpha < \kappa.$

1- Does this forcing preserve cardinals?

2-(A weaker question) Does $\kappa$ remain inaccessible in the generic extension?