I consider this issue you raised as very complex. I think for all practical purposes it would have to be considered a synthetic truth. In my view analytic truths can only be confirmed to exist for language (i.e., formal system). For example, x=1 necessarily implies that 1=x by the definition (basic set of algebraic axioms). This is a language game (algebra) and this example is analytic.
"I exist" or "I think" is a reference to me, and it is not "I" (i.e., the letter I) that is said to exist or think, rather it is I (the actual person being referenced by "I") that exists and thinks. However, how can language be said to dictate or restrict the state of anything that exists given our knowledge of language as a tool?
Rather, things that are referenced are only identified by language, and therefore they cannot be considered true by definition (i.e., true as a matter of language), but true as a matter of brute fact. Brute facts don't necessarily have to be limited by language. Rather, we expect language to be limited by brute facts.
The complexity comes in if we consider a base formal system (God) that may require the universe (brute facts) to exist. God as a 'language' would define certain events (e.g., the big bang) and as an analytical consequence other events also necessarily exist because God exists. In this case, "I exist" and "I think" could be analytical truths given the existence of the base formal system (God). Even still, for all practical purposes we cannot consider "I exist" or "I think" to be analytical truths (even *if* they really are) since we lack the knowledge to prove all this as the case (e.g., maybe the Universe is not so formal). If we had all knowledge, then maybe we would consider these to be analytical truths.
Warm regards, Harv