
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 geometric sequence $\{a_{n}\}$ where $a_{1}=2$, $a_{n+2}=4(a_{n+1}-a_{n})$, and a sequence $\{b_{n}\}$ where $b_{1}=1$, and ${b_{n+1}}=\frac{{n+1}}{n}{b_n}$.  
$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$.  
$(2)$ Let ${c_n}=\frac{1}{{4b_n^2-1}}+{a_n}$, calculate the sum of the first $n$ terms of sequence $\{c_{n}\}$ denoted as $T_{n}$.

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