
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
Points \( M, N, P, Q \) are taken on the diagonals \( D_1A, A_1B, B_1C, C_1D \) of the faces of cube \( ABCD A_1B_1C_1D_1 \) respectively, such that:

\[ D_1M: D_1A = BA_1: BN = B_1P: B_1C = DQ: DC_1 = \mu, \]

and the lines \( MN \) and \( PQ \) are mutually perpendicular. Find \( \mu \).

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