
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
The edge of a regular tetrahedron \(ABCD\) is \(a\). The vertices of the spatial quadrilateral \(A_1B_1C_1D_1\) lie on the corresponding faces of the tetrahedron ( \(A_1\) lies on the face opposite \(A\), and so on), and its sides are perpendicular to the faces of the tetrahedron: \(A_1B_1 \perp BCD\), \(B_1C_1 \perp CDA\), \(C_1D_1 \perp DAB\), and \(D_1A_1 \perp ABC\). Calculate the lengths of the sides of the quadrilateral \(A_1B_1C_1D_1\).

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