
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 $\{a_n\}$ satisfying $a_1 = 2$, and the sum of the first $n$ terms is $S_n$.
We have the recurrence relationship for $n \geq 1$,
$$
a_{n+1}= 
\begin{cases} 
pa_{n}+n-1 & \text{if }n\text{ is odd} \\
-a_{n}-2n & \text{if }n\text{ is even}
\end{cases}
$$
(Ⅰ) If the sequence $\{b_n\}$ satisfies $b_n = a_{2n} + a_{2n+1}$ ($n \geq 1$), find the sum of the first $n$ terms of sequence $\{b_n\}$, denoted as $T_n$.
(Ⅱ) If the sequence $\{c_n\}$ satisfies $c_n = a_{2n}$, determine whether $\{c_n\}$ is a geometric sequence, and explain the reason.
(Ⅲ) When $p = \frac{1}{2}$, determine whether there exists an $n \in N^*$ such that $(S_{2n+1} - 10)c_{2n} = 1$. If such an $n$ exists, find all possible values of $n$; if not, explain the reason.

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