
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 $\{a_n\}$ is a geometric sequence with $a_1=1$, the sum of the first $n$ terms is $S_n$, and $\frac{S_6}{S_3}=28$. The sum of the first $n$ terms of the sequence $\{b_n\}$ is $T_n$, and the point $(n, T_n)$ lies on the parabola $y= \frac{1}{2}x^2+ \frac{1}{2}x$.

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

$(2)$ Let $c_n=a_n\cdot b_n$, find the sum of the first $n$ terms of $\{c_n\}$, denoted as $S'_n$.

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