'gaze into' vs. 'gaze at'

Hello,
what is the difference between ‘gazing into’ and ‘gazing at’ something?
Plus: is there any difference between ‘staring at’ and ‘staring into’?
Thanks!

  1. ‘gaze into’ or ‘stare into’ – to look into something intentionally

  2. ‘gaze at’ or ‘stare at’ – to look at something intentionally