This guide provides detailed instructions on how to unlock the doors of the antechamber in the game “Blue Prince.” The antechamber has two doors: the western door (left side) and the eastern door (right side). Below are the steps to unlock each door.
WESTERN LEVER

You also need to draft the WORKSHOP
In the WORKSHOP, you need to combine the following items:
BROKEN LEVER + SLEDGEHAMMER + BATTERY PACK = POWER HAMMER
In the SECRET GARDEN, you will need to solve a puzzle. In the far end of the garden, there is a fountain with a weathervane. The arrows on the weathervane need to point toward the crescent Moon on the wall. You change the direction of the arrows by turning valves around the fountain. One valve is behind a wall. You need the POWER HAMMER to destroy the wall and use the valve.
Once all arrows point to the crescent Moon, the lever for the door will reveal itself. This seems to persist through all runs.
Some have reported that they didn’t need to do the convoluted process of getting a power hammer to break the wall. I had no such luck.
EASTERN LEVER

Behind one of the doors is the lever. The middle door on both the left and the right does not necessarily lead to the next room.