I was given two languages
$$L_1=\{0^k1^k0^m\mid k,m \in \mathbb{N}\text{ and }k < m\}$$
and
$$L_2=\{a^mb^{m+1}\}$$
and I was asked to prove whether they are context free or sensitive.
For $L_1$ : I did pumping lemma and I cannot clearly rule out the context free. So I accepted that it is and tried to do a high level description of a pda. I check the first part of the language $0^k1^k$ like this: push A for every 0 you read pop A when you read 1. If the stack is not empty reject. But I have trouble finding how to check the second part, $1^m$ with $m>k$. So now I do not know if it is indeed a context free.
For $L_2$ I followed a similar method with $L_1$ and again I stuck in the $m+1$ part.
Can someone help me? I am not proficient in grammar creation and pdas and lbas were only taught in our class in high level descriptions like the above.