
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
Let $\{a_{n}\}$ be a geometric sequence with a common ratio of $2$, and let $S_{n}$ be the sum of the first $n$ terms. Define $b_{n}=\frac{S_{n}+1}{a_{n}}$. If $\{b_{n}\}$ is a constant sequence, find:<br/>
$(1)$ The general formula for $\{a_{n}\}$;<br/>
$(2)$ Define $c_{n}=\log _{2}a_{n}$, find the sum $T_{n}$ of the first $n$ terms of $\left\{\frac{c_{n}}{a_{n}}\right\}$.

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