
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 $\{a\_n\}$, its first $n$ terms sum is $S\_n$. For all positive integers $n$, the point $P\_n(n, S\_n)$ lies on the graph of the function $f(x) = x^2 + 2x$. Let $k\_n$ be the arithmetic mean of $a\_n$ and $a_{n+1}$.

(I) Find the general term formula for the sequence $\{a\_n\}$.
(II) If $b\_n = 2^{k\_n} \cdot a\_n$, find the first $n$ terms sum $T\_n$ for the sequence $\{b\_n\}$.
(III) Let set $A = \{x \mid x = k\_n, n \in \mathbb{N}^*\}$, $B = \{x \mid x = 2a\_n, n \in \mathbb{N}^*\}$. An arbitrary term $c\_n$ of the arithmetic sequence $\{c\_n\}$ belongs to $A \cap B$, where $c\_1$ is the smallest number in $A \cap B$, and $110 < c_{10} < 115$. Find the general term formula for the sequence $\{c\_n\}$.

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