
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
Define a sequence of polynomials as follows: let \( a_{1}=3 x^{2}-x \), let \( a_{2}=3 x^{2}-7 x+3 \), and for \( n \geq 1 \), let \( a_{n+2}=\frac{5}{2} a_{n+1}-a_{n} \). As \( n \) tends to infinity, what is the limit of the sum of the roots of \( a_{n} \)?

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