
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 the sequences $\{a\_n\}$ and $\{b\_n\}$ with the general terms $a\_n= \frac{an^{2}+3}{bn^{2}-2n+2}$ and $b\_n=b-a(\frac{1}{3})^{n-1}$, where $a$ and $b$ are real constants. If $\lim\limits_{n→∞}a\_n=3$, $\lim\limits_{n→∞}b\_n=-\frac{1}{4}$, and $a$, $b$, $c$ form an arithmetic sequence, find the value of $c$.

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