You just need Stimpacks, one strong gun, ammo, armor, and a couple bobby pins. There is a perk check (Strong Back) or Survival check with Jed to increase the weight limit too, but like I said there's really no reason to take extra gear. Once inside you can either talk to Ricky for a check to get your limit increased to 100, or just talk to Jed if you have less than 75lbs. We need to make our way there and then free dog, after taking everything that isn't nailed down though. Dog is hiding away in a jail cell in the police station. If you entered through the door that puts you right in front of Dog/God, turn right and head in those locker room areas after destroying the radios. This Fallout: New Vegas walkthrough for Dead Money will help you find Collar 8 for Father Elijah. Theres some Sierra Madre Armor sitting in a locker in the police station. Fast travel to the Northern Passage (or the nearest location you've discovered) and enter the cave. Dead Money is honestly my favorite part of New Vegas, so I know it pretty well.

You can also loot the armor from Father Elijah in the Sierra Madre Casino Vault during the quest 'Heist of the Centuries'. There are a few speech checks you can make at the beginning of the DLC that allow you to increase your weight limit to 100, but you might as well unload everything so you don't have to bother with them. Found in the Brotherhood footlocker inside the Abandoned Brotherhood of Steel Bunker after completing Dead Money. I went through around 500 MF Cells with my tri-beam during this DLC. Unload everything but your Armor, Stimpacks, Bobby Pins, and favorite weapon (a Tri-Beam Laser for me), as well as plenty of ammo for it.

Head to your loot stash location (for me Doc's house one of the metal boxes). Choose the quest Happy Trails Expedition in your quest log. There are 5 achievements, all story related, and none missable as long as you make a save near the end to do the two different endings.

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