
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 an arithmetic sequence $\{ a_n \}$, where $S_n$ denotes the sum of the first $n$ terms, $S_4 = \pi$ (where $\pi$ represents the value of pi), and $a_4 = 2a_2$. Now, if we randomly select an element from the first 30 terms of this sequence, what is the probability that the cosine of this element is negative?

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