
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
The sequence $\{\,_n\}$ has a sum of the first $n$ terms denoted as $P_n$. If $3P_n=1-(\frac{1}{4})^n\ (n\in\mathbb{N}^*)$, and the sequence $\{\,_n\}$ satisfies $2\_={n+1}={n}+{n+2}\ (n\in\mathbb{N}^*)$, with $_{3}=7$ and $_{8}=22$:

1. Find the general terms $_{n}$ and $_{n}$ for the sequences $\{\,_n\}$ and $\{\,_n\}$.
2. Let the sequence $_{n}={n}{n}$. Find the sum of the first $n$ terms, denoted as $S\_n$.

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