
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
In the sequence $\{a_{n}\}$, $S_{n}$ is the sum of the first $n$ terms of the sequence $\{a_{n}\}$, and it satisfies $S_{n}+2=2a_{n}$.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$;  
$(2)$ If ${C}_{n}=\frac{{2}^{n}}{({a}_{n}-1)({a}_{n+1}-1)}$, find the sum $T_{n}$ of the first $n$ terms of the sequence $\{C_{n}\}$.

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