
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 sequence $\{a\_n\}$ with the general term formula $(a\_n = At^{n-1} + Bn + 1)$, where $A, B, t$ are constants, $t > 1$, and $n \in \mathbb{N}^*.$ The equation $(x^2 + 2x + 2)^{10} = b\_0 + b\_1(x+1) + b\_2(x+1)^2 + \cdots + b\_20(x+1)^{20},$ where $b\_i (i = 0,1,2, \cdots, 20)$ are real constants.

(1) If $A = 0, B = 1$, find the value of $\sum\_{n=1}^{10} a\_n b\_{2n}$.

(2) If $A = 1, B = 0$, and $\sum\_{n=1}^{10} (2a\_n - 2^n)b\_{2n} = 2^{11} - 2$, find the value of the real number $t$.

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