Yo and congrats on lucids!

And you are actually doing pretty well for first month! Keep practicing awareness and you will get more aware in dreams with practice, also maybe you are becoming lucid at the end of REM, which is common for first lucids since we have higher awareness at that time, in that case with practice you will be able to become aware earlier which will also prolong lucids.

As for stabilization, it is a nice and useful tool but it is not requirement for a good lucid, so instead of "Stabilize as soon as possible or else" attitude, calmly focus on your dream senses and stabilization technique, take it slowly and don"t worry about it and rather expect it to work. As for not thinking clearly, thinking about and writing/typing your goals before going to sleep and then trying to remember them after becoming lucid can be good for that, also you could try doing some basic math after becoming lucid or meditating.

As for waking up at the same time, your REM period might be ending at that time, so it's completely natural. We all briefly wake up after REM but usually we don't notice it if we are not aware enough.

Good luck with next lucids!
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