
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
For the sequence \( \left\{a_{n}\right\} \), define \( \left\{\Delta a_{n}\right\} \) as the first-order difference sequence of \( \left\{a_{n}\right\} \), where \( \Delta a_{n} = a_{n+1} - a_{n} \left(n \in \mathbf{N}^{*}\right) \). For a positive integer \( k \), define \( \left\{\Delta^{k} a_{n}\right\} \) as the \( k \)-th order difference sequence of \( \left\{a_{n}\right\} \), where \( \Delta^{k} a_{n} = \Delta^{k-1} a_{n+1} - \Delta^{k-1} a_{n} = \Delta\left(\Delta^{k-1} a_{n}\right) \).

(1) If the sequence \( \left\{a_{n}\right\} \) has the first term \( a_{1} = 1 \), and satisfies
\[ 
\Delta^{2} a_{n} - \Delta a_{n+1} + a_{n} = -2^{n},
\]

find the general term formula for the sequence \( \left\{a_{n}\right\} \);

(2) For the sequence \( \left\{a_{n}\right\} \) from (1), is there an arithmetic sequence \( \left\{b_{n}\right\} \) such that 
\[ 
b_{1} \mathrm{C}_{n}^{1} + b_{2} \mathrm{C}_{n}^{2} + \cdots + b_{n} \mathrm{C}_{n}^{n} = a_{n}
\]

holds for all positive integers \( n \in \mathbf{N}^{*} \)? If it exists, find the general term formula for the sequence \( \left\{b_{n}\right\} \); if it does not exist, explain why;

(3) Let \( c_{n} = (2n-1) b_{n} \), and define
\[ 
T_{n} = \frac{c_{1}}{a_{1}} + \frac{c_{2}}{a_{2}} + \cdots + \frac{c_{n}}{a_{n}},
\]

if \( T_{n} < M \) always holds, find the smallest positive integer value of \( M \).

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