
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 sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n = -\frac{1}{2}n^2 + kn, k \in \mathbb{N}_*$, and the maximum value of $S_n$ is 8.

(1) Determine the constant $k$ and find the expression for $a_n$.

(2) Find the sum of the first $n$ terms, denoted as $T_n$, for the sequence $\left\{\frac{9 - 2a_n}{2^n}\right\}$.

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