
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 that the geometric sequence $\{a_{n}\}$ is a strictly increasing sequence, and the product of its $3$rd, $4$th, and $5$th terms is $1000$, and after multiplying the third, fourth, and fifth terms by $4$, $3$, and $2$ respectively, the resulting three numbers form an arithmetic sequence.<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ If for any positive integer $n$, the sum of the first $n$ terms of the sequence $\{b_{n}\}$ is $S_{n}=3(1-2^{n})$, and the magnitude of the vector $(a_{n}$,$b_{n})$ is $t_{n}$, find the sum of the first $n$ terms of the sequence $\{t_{n}\}$.

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