
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
From the 8 vertices of the cube ABCD-A1B1C1D1, any 4 different vertices taken may form:

(1) The 4 vertices of a rectangle;

(2) The 4 vertices of a tetrahedron, each of whose faces is an equilateral triangle;

(3) The 4 vertices of a tetrahedron, each of whose faces is a right triangle;

(4) The 4 vertices of a tetrahedron, three of whose faces are isosceles right triangles and one of whose faces is an equilateral triangle.

The number of correct conclusions among these is $\boxed{\text{4}}$.

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