Day Seventy Eight is a part of the story in Firewatch.
Talk to Delilah, then head towards Cave 452, just North of your lookout but before Thunder Canyon. Get in the cave, and unlock the door. Keep moving forward till you get to a drop you can't do, turn left, and find the loose rocks. Break on through to the other side.
After you exit the cave, you'll find the Kid's Hideout. Pick up everything here, and talk to Delilah if you want to report any of it. Grab the homemade sign. Take the pitons underneath it.
Continue to follow the path North, and you'll find the cracked rock. Drive the piton into the crack and you can rappel down, then head west to the trail, then south again to get back to Cave 452. Use the key, and this time you can use that piton to go down the drop you couldn't get down earlier. Explore, then head back out when the objective updates for the placard for Day Seventy Nine.
|Day One||Day Two||Day Three||Day Nine|
|Day Fifteen||Day Thirty Three||Day Thirty Four||Day Sixty Four|
|Day Seventy Six||Day Seventy Seven||Day Seventy Eight||Day Seventy Nine|