
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
In the sequence $\{a_n\}$, $a_n > 0$, $S_n$ is the sum of the first $n$ terms, and the vector $\overrightarrow{AB}=(S_n,p^2-a_n)$, $\overrightarrow{CD}=(1,p-1)$, and $\overrightarrow{AB} \parallel \overrightarrow{CD}$, where $p > 0$ and $p \neq 1$.
1. Find the general formula for the sequence $\{a_n\}$.
2. If $p= \frac{1}{2}$, the sequence $\{b_n\}$ satisfies that for any $n \in \mathbb{N}^*$, $b_1a_n+b_2a_{n-1}+\ldots+b_na_1=2^n- \frac{1}{2}n-1$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, $T_n$.

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