
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
We define: For any real number $A$, if there exists a sequence $\{a_n\}$ and a real number $x$ ($x \neq 0$), such that $A = a_1 + a_2x + a_3x^2 + \ldots + a_nx^{n-1}$, then we say that the number $A$ can be represented in base $x$, denoted as: $A= \overline{x(a_1)(a_2)(a_3)\ldots(a_{n-1})(a_n)}$. For example: $A= \overline{2(-1)(3)(-2)(1)}$, this means $A$ is a number in base $2$, and $A = -1 + 3 \times 2 + (-2) \times 2^2 + 1 \times 2^3 = 5$.
1. Given $m = (1-2x)(1+3x^2)$ (where $x \neq 0$), try to represent $m$ in the base $x$ notation.
2. If the sequence $\{a_n\}$ satisfies $a_1 = 2$, $a_{k+1} = \frac{1}{1-a_k}$, $k \in \mathbb{N}^*$, and $b_n = \overline{2(a_1)(a_2)(a_3)\ldots(a_{3n-2})(a_{3n-1})(a_{3n})}$ ($n \in \mathbb{N}^*$), is there any real constants $p$ and $q$ such that for any $n \in \mathbb{N}^*$, $b_n = p \cdot 8^n + q$ always holds? If yes, find $p$ and $q$; if no, explain why.
3. If the constant $t$ satisfies $t \neq 0$ and $t > -1$, and $d_n = \overline{2( C_{ n }^{ 1 })( C_{ n }^{ 2 })( C_{ n }^{ 3 })\ldots( C_{ n }^{ n-1 })( C_{ n }^{ n })}$, find $\lim_{n \to \infty} \frac{d_n}{d_{n+1}}$.

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