
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 arithmetic sequence $\{a_n\}$, $a_1=1$, and the sum of the first $n$ terms $S_n$ satisfies the condition $\dfrac{S_{2n}}{S_{n}}= \dfrac{4n+2}{n+1},n=1,2,...$,

(Ⅰ) Find the general formula for the sequence $\{a_n\}$;

(Ⅱ) Let $b_n=a_np^{a_n} \left( P > 0 \right)$, 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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