
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
In the cube \\(ABCD-A_1B_1C_1D_1\\), \\(P,Q\\) are the centers of the adjacent side faces \\(BCC_1B_1\\) and \\(CDD_1C_1\\) respectively. Find the cosine of the angle formed by the skew lines \\(A_1P\\) and \\(B_1Q\\).

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