
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 an arithmetic sequence $\{a_{n}\}$, take one number from each of the first, second, and third rows of the table below, respectively, as $a_{1}$, $a_{2}$, $a_{4}$, and ensure that any two numbers in $a_{1}$, $a_{2}$, $a_{4}$ are not in the same column. The first three terms of a geometric sequence $\{b_{n}\}$ with a common ratio greater than $1$ are exactly three terms from the first $5$ terms of the sequence $\{a_{n}\}$.

|         | Column 1 | Column 2 | Column 3 |
|---------|----------|----------|----------|
| Row 1   | $8$      | $0$      | $2$      |
| Row 2   | $7$      | $4$      | $3$      |
| Row 3   | $9$      | $12$     | $4$      |

$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;  
$(2)$ Let ${c_n}=\frac{{{a_n}{b_n}}}{{{a_{n+1}}{a_{n+2}}}$, 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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