
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 point P is the midpoint of the diagonal BD₁ of the cube ABCD-A₁B₁C₁D₁, plane α passes through point P and is perpendicular to line BD₁, and plane α intersects plane ABCD at line m, calculate the cosine of the angle formed by line m and line A₁C.

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