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After a short introductory cut-scene with Anna, speak with the boy Josh who's walking around the docks. For a morsel of food (soy will do), he'll tell you (as a matter of fact) about the secret entrance near the soda machine, and also give you the code. Alternatively you can use a LAM to blow up the soda machine rather than wasting the lockpicks on the nearby security panel box. If you jump into the water here by the boat you'll find two crates with useful items.
You'll have no primary goals and one secondary goal at this time: locate the ambrosia barrel in Castle Clinton.
Enter the facility from behind the soda machine and go straight into another room for supplies, then back out and downstairs.
There's a lone guard patrolling this area, so look out. Wait for him to walk away and slip all the way down. Take an immediate right, find the multitool on the floor, and enter the unlocked ventilation shaft to the right of the tool.
Crawl through the shaft past a few possible off-shoots leading to openings until you reach the very end. The off-shoots lead only to the same long walkway that's being patrolled by guards. You'll hear them walking along side you as you crawl through. You'll also know you're at the end if you see a long corridor through the end grate.
Be careful, as a guard with a flamethrower likes to stroll down this hallway into the room at the end, and then back out. Don't waste your time with him unless you want the weapon. Instead, once he's back out of the room, crawl out into the hallway and travel down the corridor to the far room, where a couple of useful items await and another guard will eventually find his way into.
This room has another exit to its left at a right angle into another shorter corridor where a guard patrols. He'll come as far as this room, so wait until he does, and stun/prod/kill him (if you wish) when he enters. I usually choose during these beginning stages to use the prod on guards' legs or lower back so as not to kill. Be careful of the guard coming into the room from the corridor you just came from. He will see you if you wait that long. Fortunately, it's timed so that you can wait for the next guard, deal with him, and exit before the first guard walks in.
Past the left corridor you'll enter a smallish square room with a stairway exit at the far end, and an exit along the left wall and stacked crates. There's also a security camera along the top right wall obscured by the crates. Slide along the right edge and then under the camera when it moves away from you. While under it, when the camera moves away from you to its left, make your way out along the north edge of the room, bypassing the stairway into the next room, and you'll see a larger room with water and a low wooden dock. On the other side of a big crate on the dock is your first secondary goal, to find the ambrosia.
Another secondary goal will be added this time: report to agent Navarre outside. There are two approaches you can take. You can follow the stairways up and then up again to the castle courtyard that's crawling with NSF troops, or go for a more subtle approach: backtrack back out of the soda machine. If you want to maintain your pacificity, I'd suggest this alternate route... as if they spot you in the courtyard, you'll begin a small battle with Anna, UNATCO and the NSF outside and you'll be branded a killer. But before backtracking to the soda machine, let's head up the first stairway we just passed in the other room.
When you get upstairs, enter the small hallway, turn left and you'll find another small stairway leading down to a locked door. Unlock it with lockpicks, and enter to find some nice supplies, an aqua lung AUGMENTATION CANISTER
and weapon mod within the safe. Simply use the nanokey you find on the computer console.
If you still want to go to the courtyard, simply head up the small stairway, and then around the hallway to another stairway. At its top is a button which opens the upper covering to yet another stairway here. At its top are a couple locked doors you can open with the nanokey you probably picked up from the room mentioned just above.
Backtrack to the soda machine. It's not very difficult... you simply need to look out for two guards. Once outside, head around to the west side of Battery Park to where the citadel walls are. Around the north side of this large structure you'll find a few UNATCO troops guarding the open entrance to the courtyard (described above). At some point around the building's circumference, Anna will come running to you and update you with your first primary goal: free the hostages in the subway station to the east of Castle Clinton.
There are a few supply crates strewn about the western perimeter of Battery Park if you want to check them. Now head north and east to shack central, where a small battle between the NSF and UNATCO is being waged. Do not help out if you want to remain peaceful. Instead, wait it out, breaking open the few supply crates around the edge of the area. The UNATCO troops will win, with the possibility of a few NSF stragglers inside a shack.
Once the area is clear, there are a few ways into the Subway. The most direct approach is ill-advised. That's the entrance to the subway station where several NSF wait downstairs. You'll have to kill them and bypass the laser field with an EMP grenade if you want to take that approach. We'll take the northern-most shack entrance. Inside the shack is an unlocked grate with steam coming out of it.
Climb down the grate and take the very first shaft on your way down the ladder. You could keep going down if you wanted to the subway floor but we're going to get at the NSF from above for now.
Take this shaft to its end, take a left and immediate right into another shaft, and follow it until its dead end where you'll find a grate looking over a part of the subway station, likely with a couple NSF below. Alex Jacobson will warn you about booby traps. Attempt to get the NSF to see or hear you by firing a shot into the wall while over the grate. When they do, you'll find about 5 to 7 NSF congregating below, running this way and that and shooting at you. Now's your chance to open the grate and toss a gas grenade straight down. When they start reeling and coughing, shoot them with tranquilizer darts to knock them out. When all of them are out, Alex will com you, "Good Job...".
Well, that takes care of the NSF guarding the hostages. Now to get down there. Backtrack to the ladder with the steam that goes up to the shack. It may be a little difficult to find but use your light (F12) and you should be okay, even if you get temporarily lost. Once on the ladder, I'd suggest going back up and out to the subway entrance, and walk down the stairs to break open the couple crates down there. You can of course then also toss an EMP grenade at the laser field nearby and walk through that, but why waste one? Once back at the ladder with steam, this time go all the way down. At the bottom of the ladder you'll find another shaft. Take the first grate exit you find out into the subway station (there are several), and walk around to collect things off the NSF and inside crates. Also receive thanks from the hostages.
Open the subway doors using the panels on the train, and the hostages will run inside. Once their in, step inside to travel to Hell's Kitchen where you're supposed to meet up with Paul.