
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
Let  $ d_n$  be the determinant of the  $ n\times n$  matrix whose entries, from left to right and then from top to bottom, are  $ \cos 1,\cos 2,\dots,\cos n^2.$  (For example,  $ d_3 \equal{} \begin{vmatrix}\cos 1 & \cos2 & \cos3 
\cos4 & \cos5 & \cos 6 
\cos7 & \cos8 & \cos 9\end{vmatrix}.$  The argument of   $ \cos$   is always in radians, not degrees.)

Evaluate  $ \lim_{n\to\infty}d_n.$ 

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