
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]


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The triangle \( ACH \) is defined by three vertices of a cube, as shown in the diagram. The height of this triangle relative to side \( CH \) is \( 12 \) cm.

Calculate the area of triangle \( ACH \) and the length of an edge of the corresponding cube.

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