
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 sequences $\{a_n\}$, $\{b_n\}$, and $\{c_n\}$ satisfy $a_1=a$, $b_1=1$, $c_1=3$, and for any $n\in\mathbb{N}^*$, we have $b_{n+1}= \frac{a_n + c_n}{2}$ and $c_{n+1}= \frac{a_n + b_n}{2}$.
(1) Find the general formula for the sequence $\{c_n-b_n\}$.
(2) If the sequence $\{a_n\}$ and $\{b_n+c_n\}$ are both constant sequences, find the value of the real number $a$.
(3) If the sequence $\{a_n\}$ is a geometric progression with the common ratio $a$, let the sum of the first $n$ terms of sequences $\{b_n\}$ and $\{c_n\}$ be $S_n$ and $T_n$ respectively, and let $M_n=2S_{n+1}-T_n$. Determine the range of values for $a$ such that $M_n < \frac{5}{2}$ holds for any $n\in\mathbb{N}^*$.

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