
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 $\{a_{n}\}$, $a_{1}=2$ and ${a}_{n}^{2}={a}_{n+1}$ $(n\in{N}^{*})$.  
$(1)$ Find the sum $S_{n}$ of the first $n$ terms of the sequence $\{\log _{2}a_{n}\}$.  
$(2)$ Let $b_{n}$ be the number of $t(t\in N^{*})$ satisfying the inequality $a_{n}\leqslant t\leqslant a_{n+1}$. The sum of the first $n$ terms of the sequence $\{b_{n}\}$ is $T_{n}$. Find $\log _{2}(T_{8}-6)$.

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