Suppose that $\{a_n\}$ is an arithmetic sequence with  
$$a_1+a_2+\cdots+a_{100}=100 ~\text{and}~ a_{101}+a_{102}+\cdots+a_{200}=200.$$ 
If the answer is $x$, what is the value of $100x$?

Let's think step by step and output the final answer within \boxed{}.
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