
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 the sequence ${a_n}$ satisfying $2a_{n+1}=a_{n}+a_{n+2}$ for all $n\in\mathbb{N}^*$, with $a_5=5$ and the sum of its first $n$ terms denoted as $S_n$ and $S_7=28$.

(1) Find the sum of the first $n$ terms, denoted as $T_n$, of the sequence ${\frac{1}{S_n}}$.
(2) If sequence ${b_n}$ satisfies $b_{1}=1$ and $b_{n+1}=b_{n}+q^{a_n}$ where $q>0$, find the general term formula for ${b_n}$ and compare the size of $b_n⋅b_{n+2}$ and $b_{n+1}^2$.

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