
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 that the sum of the first $n$ terms of a geometric sequence $\{a\_n\}$ satisfies $S\_n=1-A\cdot3^{n}$, and the sequence $\{b\_n\}$ is an increasing sequence where $b\_n=An^{2}+Bn$, determine the value of $A=$ \_\_\_\_\_\_, and the range of values for $B=$ \_\_\_\_\_\_.

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