
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 $\{a_n\}$ whose sum of the first $n$ terms is denoted by $S_n$, and $S_n = n^2 - 4n + 4$ (where $n$ is a positive integer),
1. Find the general term formula for the sequence $\{a_n\}$.
2. For sequence $\{b_n\}$, let $b_n = \begin{cases} 
1, & n=1 \\
\frac{a_n + 5}{2}, & n \geq 2 
\end{cases}$, and let $T_n = b_1 \cdot 2^1 + b_2 \cdot 2^2 + b_3 \cdot 2^3 + \dots + b_n \cdot 2^n$. Find $T_n$.
3. Consider a sequence $\{c_n\}$, where all terms are nonzero, and define the number of positive integers $i$ such that $c_i c_{i+1} < 0$ as the number of sign changes in the sequence $\{c_n\}$. Let $c_n = 1 - \frac{a}{a_n}$ (where $n$ is a positive integer). Find the number of sign changes in the sequence $\{c_n\}$.

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