2 added 148 characters in body

Let $\kappa$ be an inaccessible cardinal. Is there any forcing notion $P$ with the following properties:

1-$P$ preserves GCH and the strong inaccessibility of $\kappa$,

2-$P$ adds a subset of $\kappa$ of size $\kappa$,

3-conditions in

3-$P$ is the $< \aleph_1-$support product of some forcing notions $P_{\alpha}, \alpha < \kappa.$

4- The generic filter for $P$ are at most countablecan be reconstructed from the subset of $\kappa$ added in 2.

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# Adding large sets by countable conditions preserving the GCH

Let $\kappa$ be an inaccessible cardinal. Is there any forcing notion $P$ with the following properties:

1-$P$ preserves GCH and the strong inaccessibility of $\kappa$,

2-$P$ adds a subset of $\kappa$ of size $\kappa$,

3-conditions in $P$ are at most countable.