
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 \( ABC - A_1B_1C_1 \) is a right prism with \(\angle BAC = 90^\circ\), points \( D_1 \) and \( F_1 \) are the midpoints of \( A_1B_1 \) and \( B_1C_1 \), respectively. If \( AB = CA = AA_1 \), find the cosine of the angle between \( BD_1 \) and \( CF_1 \).

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