
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 that the sequence $\{a_n\}$ is an arithmetic sequence, and $S_n$ is the sum of the first $n$ terms of $\{a_n\}$, with $a_{10}=19$ and $S_{10}=100$. For the sequence $\{b_n\}$, it holds that $b_1 \cdot b_2 \cdot b_3 \ldots b_{n-1} \cdot b_n = a_n + 2$ for any $n \in \mathbb{N}^*$.

(Ⅰ) Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$;

(Ⅱ) Let $c_n = (-1)^n \frac{4n \cdot b_n}{(2n+1)^2}$, find the sum of the first $n$ terms of the sequence $\{c_n\}$, denoted as $T_n$.

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