
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 it satisfies $S_n=2a_n-2$ ($n\in\mathbb{N}^*$), and the sequence $\{b_n\}$ is an arithmetic sequence, satisfying $b_2=a_1$, $b_8=a_3$.  
$(I)$ Find the general formulas for sequences $\{a_n\}$ and $\{b_n\}$.  
$(II)$ Let $c_n=1-(-1)^{n+1}a_n$, and the solution set of the inequality $c_k \geqslant 4097$ ($1\leqslant k\leqslant 100$, $k\in\mathbb{N}^*$) is $M$. Find the sum $S$ of all $a_k+b_k$ ($k\in M$).

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