The language $ L$ defined as: $ L = \{wtw^R | w,t \in \{0,1\}^\star \text{ and } |w| = |t| \}$

It looks like the language can be “pump” by context free pumping lemma, but pumping lemma doesn’t prove the language is context free. So I’m thinking to build a PDA that recognized it. But I’m stuck on constructing the PDA, my initial thought is using nondeterminism to guess the string $ w$ and $ t$ , then compare the values after $ t$ which is string $ w^R$ . But still hard for me to come up a whole construction at this moment, any suggestions?