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

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