
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 $\{a_n\}$, it is known that $a_3=5$, and $a_1$, $a_2$, $a_5$ form an increasing geometric sequence.
(Ⅰ) Find the general formula for the sequence $\{a_n\}$;
(Ⅱ) If the general formula for the sequence $\{b_n\}$ is $b_n= \begin{cases} a_{ \frac {n+1}{2}}, & n=2k-1 \\ 2^{ \frac {n}{2}-1}, & n=2k \end{cases} (k\in\mathbb{N}^*)$, find the sum of the first $n$ terms of the sequence $\{b_n\}$.

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