
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
Given that the sequence $\{a_n\}$ is an arithmetic sequence with a positive common difference, and the sequence $\{b_n\}$ is a geometric sequence, with $a_1=1$, $b_1=2$, $b_2-a_2=1$, and $a_3+b_3=13$,

(Ⅰ) Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$.

(Ⅱ) Let $c_n=a_n\cdot b_n$, find the sum of the first $n$ terms of the sequence $\{c_n\}$, denoted as $T_n$.

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