I think the ultimate way to access a doc would be through a widget on android. Have a quick thought, swipe to a page on your home screen, unload the thought onto the widget, and it syncs with the doc on the board. This would probably be most useful with the regular text based doc, and just having it sync with a doc on your board, but this could probably expand into having a few different widget types that let you access several different doc types.
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