
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 corner of a unit cube is cut off such that the cut runs through the three vertices adjacent to the vertex of the chosen corner, forming a new face that is an equilateral triangle. If the side length of the unit cube is increased to 2 units, what is the height of the modified cube when the freshly cut face is placed on a table?

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