
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 a sequence $\{a_n\}$ with the general term formula $a_n = A t^{n-1} + Bn + 1$, where $A$, $B$, and $t$ are constants with $t > 1$ and $n \in \mathbb{N}^*$. The following equation holds: $(x^2 + x + 2)^{10} = b_0 + b_1 (x + 1) + b_2 (x + 1)^2 + \ldots + b_{20} (x + 1)^{20}$, where $b_i$ ($i = 0, 1, 2, \ldots, 20$) are real constants.
1. If $A = 0$, $B = 1$, find the value of $\sum\limits_{n = 1}^{10} a_n b_{2n}$;
2. If $A = 1$, $B = 0$, and $\sum\limits_{n = 1}^{10} (2 a_n - 2^n) b_{2n} = 2^{11} - 2$, find the value of the real number $t$.

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