
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 that \( A_{1} B_{1} C_{1}-A B C \) is a right triangular prism, with \( \angle BCA = 90^{\circ} \). Points \( D_{1} \) and \( F_{1} \) are the midpoints of \( A_{1} B_{1} \) and \( A_{1} C_{1} \), respectively, find the cosine of the angle between \( BD_{1} \) and \( AF_{1} \).

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