Saw V

Convicted murderer Seth Baxter wakes up chained to a table beneath a pendulum blade. A videotape informs him that he can release himself by crushing his hands between two presses. He does so, but the blade still swings down and cuts him in half as someone watches through a hole in the wall.
In the meatpacking plant, FBI Agent Peter Strahm guns down Jeff Denlon and is locked in the sickroom. He discovers a hidden exit and a microcassette recorder, which urges him to stay in the sickroom, but he ignores it and is attacked in the tunnel by a pig-masked figure. He wakes up with his head sealed in a box which quickly fills with water, but survives by performing a tracheotomy with his pen. Outside, Detective Mark Hoffman delivers Corbett Denlon to the police and claims they are the only survivors, but is shocked when Strahm is brought out alive as well.
During a press conference, Hoffman is promoted to detective lieutenant and credited with closing the case. He finds a note in his office that reads "I know who you are" and learns of Lindsey Perez's death while taking Strahm's phone. At the hospital, Strahm tells Hoffman that Perez's last words were his name, and questions how he escaped the plant. Strahm is also visited by his boss, Dan Erickson, who puts him on medical leave. Suspicious of Hoffman and determined to piece together his involvement with Jigsaw, Strahm takes case files of past victims and begins researching on his own.
In an underground room, five people awaken with collars connected to mounted blades locked around their necks, the keys to which are in glass boxes across the room. A videotape claims that all five of them are connected and implores them to "do the opposite" of their instincts throughout the game. Mallick, an arsonist, activates the timer and everyone reaches their keys except Ashley, a former fire inspector, who is decapitated when the collars retract. In the next room, they learn that the overhead jars contain keys to three bomb shelters. Charles, an investigative journalist, strikes Mallick and begins smashing the jars one by one. Brit, a real estate vice president, and Luba, a city planner, each find a key and Charles takes Mallick's, only to be struck by Luba and left to die when the bomb detonates. In the third room, five short cables must be connected to a full bathtub to unlock the door. Luba attacks Mallick to use his body to close the circuits, but Brit stabs her and they use hers instead. In the fourth room, they find a machine fitted with circular saws and a beaker requiring ten pints of blood to open the door. Mallick notes the five armholes and they realize that all five of them were meant to work together to survive the game. Brit also works out their connection: they were all involved with a building fire that killed eight people. With no other options, they saw their arms to provide the blood.
Meanwhile, Strahm learns that Baxter killed Hoffman's sister, and was released from prison on a technicality. As revenge, Hoffman later abducted and killed Baxter, using the pendulum trap to frame Jigsaw. He was later abducted by John Kramer and blackmailed Hoffman into working with him. They set up most of the games together, with Hoffman planting Lawrence Gordon's penlight and providing the police files for the nerve gas house victims. Strahm realizes in the end that everyone was meant to die in the plant except for Corbett and Hoffman, who would become a hero. His activities soon draw Erickson's worry, which is further fueled when Jill, who received a box and a videotape from Kramer's will, approaches Erickson and claims Strahm is stalking her. After Hoffman calls Erickson to tell him about Strahm's theory of a second accomplice, Erickson puts a tracker on Strahm's phone.
Erickson follows the tracker to the observation room and finds the phone and his own personnel file, planted by Hoffman. He also finds Brit, who was managed to crawl from the fourth room after Mallick passed out from blood loss. Erickson calls for medical attention for the victims, and puts out an all-points bulletin on Strahm, who follows Hoffman to the renovated nerve gas house, and finds an underground room containing a clear box filled with broken glass. Hoffman's tape urges Strahm to enter the box, but he stops it short and ambushes Hoffman. After a brief struggle he seals Hoffman in the box, which seals the room's exit. Hoffman indicates the tape, which warns Strahm that if he doesn't enter the box he will disappear and inherit the Jigsaw legacy. The box is safely lowered into the floor as the walls close in on Strahm, who unsuccessfully attempts to escape through the ceiling grid, and is crushed to death.

Question: Who works out their connection of the five people?
Answer: Brit