
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
Given a cube \( ABCD-A_{1}B_{1}C_{1}D_{1} \), find the value of \( \cos \theta \), where \( \theta \) is the angle between the planes \( AB_{1}D_{1} \) and \( A_{1}BD \), with \( 0^{\circ} \leq \theta \leq 90^{\circ} \).

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