
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 a sequence $\{a_n\}$ that satisfies $2a_{n+1} = a_n + a_{n+2}$ for $n \in \mathbb{N}^*$, the sum of the first $n$ terms is denoted as $S_n$, and it is given that $a_5 = 5$ and $S_7 = 28$.  
(1) Find the sum of the first $n$ terms of the sequence $\left\{\frac{1}{S_n}\right\}$, denoted as $T_n$.  
(2) Suppose another 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 the sequence $\{b_n\}$, and compare the magnitude of $b_n \cdot b_{n+2}$ with $b_{n+1}^2$.

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