
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 $\{ \triangle ABC \}$ with the first term $a_{1}=1$ and common difference $d$, and the sequence $\{2^{a_n}\}$ is a geometric sequence with a common ratio of $4$,

 $(1)$ Find $d$;  
 $(2)$ Find the general formula for the $n$-th term $a_{n}$ and the sum of the first $n$ terms $S_{n}$;  
 $(3)$ Find the sum of the first $n$ terms $T_{n}$ of the sequence $\left\{ \frac{1}{{a}_{n}{a}_{n+1}} \right\}$

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