Also, so far Israel have not investigated how many of the civilians were killed by friendly fire. So
You stated that Isreal exaggerated and lied about 1200 civilians being killed. That confirm 1137 were killed.
That alone was enough to go after Hamas I don’t think they needed much more of a reason
yes they said Hamas killed 1200 CIVILIANS plus lots of other exaggerations which have been proven false, but are still used in official narratives
However 1137 People were killed - those people include Hamas attacker, IDF soldiers and Civilians - so far there has been no clarification on how many of the civilians killed, were murdered by Hamas or by "Friendly fire" There is video evidence and interviews from survivors to suggest many were killed by the IDF.
There were also an official narrative that Hamas deliberately targeted the musical festival. This was proven false as the musical festival was due to end the day before, with an extension only given 3 days before the attack. Also, journalists who watched the heavily produced 40minute video (used to convince journalists and influencers to support Israel) stated that the GoPro footage recovered showed Hamas terrorists surprised by the event and looking for IDF.
So, Yes - they exaggerated and lied.
And as I also said, but you seem to have ignored
"The stupid thing is the truth was bad enough, but they exaggerated it to justify the response they wanted"
this is why blindly believing what comes from Official Israeli channels is as stupid as believing what comes from Official Hamas channels