There is no required post order, no. The only thing you need to be aware of is that in the event that you haven't replied IC in a while, either myself, or Nallore will dictate what your character does, just to keep said character in the action and involved for when you return. Our dictations will be minor, without having and major lasting impact in your character, and will be done the best of our abilities. A good example would be if the Fireteam were moving into a location, and everyone but your character had been posted for; myself or Nallore would simply make a little note in one of our posts that your character followed suit. This is so that if you're busy, or if you're away, you don't need your character to miss out on things, and when you come back, you'll be able to just hop right back into control.
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And yes, all weapons of the Halo Universe that were in use up the the year 2550, will be available to your characters. However, your characters will be limited to the number of weapons your character can carry, just as in the Halo Games. Generally One primary weapon, and one secondary.