
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 $\left\{a_n\right\}$, the sum of its first $n$ terms is $S_n$, where $S_n=2a_n-2$.

$(1)$ Find the general formula for the sequence $\left\{a_n\right\}$.

$(2)$ Let $b_n=\log_2(a_n)$, $c_n= \frac{1}{b_nb_{n+1}}$, and denote the sum of the first $n$ terms of the sequence $\left\{c_n\right\}$ as $T_n$. If for $n\in\mathbb{N}^*$, $T_n\leqslant k(n+4)$ always holds, find the range of values for the real number $k$.

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