
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
A unit cube has a corner chopped off such that the cut runs through the three vertices adjacent to the chosen corner vertex. The freshly-cut face, which is an equilateral triangle, is placed on a table. The side length of this triangle is $\sqrt{2}$. The volume of the chopped-off portion (a pyramid) is $1/6$ of the cube's volume. What is the height of the remaining part of the cube?

Present the answer in LaTex format: \boxed{Your answer}
user