Eutoum Shrine is near Goflam's Secret Hot Spring in the northern part of the Hebra Mountains.
Shrine Location
Eutoum Shrine is at -3506, 3570, 0387 in the northern part of the Hebra Mountains, near Goflam's Secret Hot Spring.
Proving Grounds: Infiltration
Take the weapons and shield from the rack up ahead, then slowly go toward the column that has a lamp on it. You can hide behind this column and watch for the Soldier Construct to turn its back to you, then try to crouch and sneak up on it to do a sneakstrike. You can also hide behind the smaller lamp next to the Construct while you wait. The Construct will turn left, then straight ahead, then right, so watch it until you know the timing and can sneak up on it at the right time. After you defeat it, fuse its horn to your weapon.
You can move the spikes behind where this Construct was, but for now, use Ascend to get on top of the ledge along the wall. Now that you have a bow and some arrows, you can go around the right side and try to shoot the Construct that is patrolling on that side.
On the ledge, at the sides, there are laser beams. One side has laser beams that you can crouch to get past. If the laser beam hits you, an alarm will sound, alerting the Constructs to your presence. You can hide somewhere to make them stop coming after you.
There are three Constructs in the walled area. One of them will shoot shock arrows at you. You can try to drop things on them from above. You can also get arrows by breaking the barrels and crates around the area.
In the corner, there is a Flame Emitter that will shoot flames at you when you are nearby. However, you can get close enough to move it with Ultrahand. It is attached to a spike, allowing you to place it in the floor wherever you wish. Place it near a Construct and it will shoot its flames at the Construct instead.
Shrine Reward
After you defeat all the Constructs, the gate will open. Go through open the chest to get a Sapphire. Then check on the dragon ring to get a Light of Blessing and complete the shrine.