
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 a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and $S_n=2a_n-2$ for $n=1,2,3\cdots$, and in the sequence $\{b_n\}$, $b_1=1$, the point $P(b_n, b_{n+1})$ lies on the line $x-y+2=0$.

$(1)$ Find the general terms $a_n$ and $b_n$ for the sequences $\{a_n\}$ and $\{b_n\}$ respectively;

$(2)$ Let $H_n= \frac{1}{b_1b_2}+ \frac{1}{b_2b_3}+\ldots+ \frac{1}{b_{n-1}b_n}$, find the smallest positive integer $m$ such that $H_n < \frac{m}{30}$ holds for all $n\in\mathbb{N}^*$;

$(3)$ Let $c_n=a_n\cdot b_n$, find the sum of the first $n$ terms $T_n$ of the sequence $\{c_n\}$.

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