
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
Let $\{a_n\}$ be an arithmetic sequence with the sum of its first $n$ terms denoted by $S_n$ $(n \in \mathbb{N}^*)$; $\{b_n\}$ is a geometric sequence with a common ratio greater than $0$, and the sum of its first $n$ terms is denoted by $T_n$ $(n \in \mathbb{N}^*)$. Given that $b_1=1$, $b_3=b_2+2$, $b_4=a_3+a_5$, and $b_5=a_4+2a_6$,
(Ⅰ) Find $S_n$ and $T_n$;
(Ⅱ) If $S_n + (T_1 + T_2 + \ldots + T_n) = a_n + 4b_n$, find the value of the positive integer $n$.

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