
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 an arithmetic sequence $\{a_n\}$ with common difference $d \neq 0$, the sum of the first $n$ terms is $S_n$, and it is known that $S_5=35$, and $a_1$, $a_4$, $a_{13}$ form a geometric sequence.

$(1)$ Find the general formula for the sequence $\{a_n\}$;

$(2)$ Given that the sequence $\left\{\dfrac{b_n}{a_n}\right\}$ is a geometric sequence with the first term as $1$ and common ratio as $2$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, $T_n$.

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