
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 and $\{b_n\}$ be a geometric sequence with the sum of the first $n$ terms being $S_n$. Given that $a_1=-1$, $b_1=1$, $a_2+b_2=2$, and $a_3+b_3=5$.
(Ⅰ) Find the general formula for the $n$-th term of the sequence $\{b_n\}$ and the sum of its first $n$ terms $S_n$;
(Ⅱ) Let $c_n=a_n+\log_2 b_n$, find the sum $T_n$ of the first $n$ terms of the sequence $\{c_n\}$.

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