I submit your metaphor has a very strong bias. "The doors" we examine do not look the same. Further, we aren't in an empty room, we are in a world that, whatever faith we're considering, has to exist in. Finally, my particular faith, the adherent believes they have been fundamentally changed such that their perception of "the doors" is also altered necessarily making them appear different.
So, will I consider other faiths? Absolutely.
Can I "reason" through things? Sort of. There are presuppositions that preclude two people from using the same evidence in the same way. As an example, does the same biological mechanism for locomotion indicate a common creator or a common ancestor? There's a lot of presupposing that goes one.
I recommend Cornelius Van Til's though (perhaps digested by someone more contemporary, like a John Frame) for more on this. Van Til is Christian, so that's the perspective that you're going to encounter, but I think it's helpful toward productive, respectful dialog to know about presuppositions going into any engagement.