Cloudbuilt's Core story is about having life give you circumstances and you making decisions about how to evaluate them. It is a deep for of understanding who you are personally and how you percieve yourself. The Fog Core is an extension of this idea. It can serve 2 purposes that we've found, first is a way to help get a pacifist achievement. The second purpose and the reason why you are reading this guide is that it unlocks 3 of the most combat heavy levels in the game- The Storm, The Night, and the Beyond.

The narrator to the fog core is a serpent luring you into a space embodied of your aggressive behaviour. The Fog Core when used in the pacifist way blocks off your path to the Ash and Hawthorn branches. (Hint 1) While the purpose of blocking the defiance levels is not clear, the purpose of blocking the fog core branches is the first clue on how to use the fog core. The three levels in particular which are tied to the fog core are Redeployment-The storm, My Regrets-The night, and Questions-The beyond.

The second key part of the puzzle is that the levels are labeled in supercloudbuilt. There are green(incomplete) and white(complete) markers on the bottoms of the doors. This is interpreted as the hospital environment gives you the hints as to which levels to look at. (Hint 2). The last hint is that you can use the fog core to enter a level and leave it to search another one within the branch behind the enclosed walls of ash and hawthorn. (Hint 3)

Preparation-
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To use the fog core to unlock the levels its pretty simple. I'll give you my order of events to complete them. First, I'd gather as many items as you can around the map until you have a full inventory. Second, I'd get the fog core from the another way level. Third, I would backup my save file in the appdata/roaming/supercloudbuilt folder. Now with this prepartion done.
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I would.
1. After creating your backup, go into a branch this is going to be locked, for example, the Ash Branch.
2. Stand in front of the door of the level you are trying to unlock I would use the fog core, (redeployment would be my choice because the fog core level the storm is much much easier than the Night).
3. Use the fog core.
4. Enter the level.
5. Leave the level through the entrance you just entered through.
6. In the hallway you just were, the level should appear.
7. Complete the level (preferably the storm)
8. After respawning in the hospital, backup your save, then go into questions.
9. In question find the hidden door which is about 1/3rd of the way into the level under two walking aractoids with the red no energy field on a fence requiring you to drop down inside of it.
10. Enter the door into the fog core level select.
11. After entering the door, complete the other level which is in the beginning part, likely, the NIght.
12. In the night, use every powerup you can to complete the level. It is the hardest level in the game.
13. After completing the night, go back into questions and enter the level select again.
14. Complete the last level and enjoy the secret ending.
15. Compete on the leaderboards for the unclocked 3 levels in ranked.
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Hope this helps!
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Shoutout to shady fountain ElleAwyn, wobs23, Iijil, Rusty, Chi-pa-pa, Kalifen, and the supercloudbuilt reddit /r/cloudbuilt for helping to figure all this out.