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How to find fixed points for DES weak keys

Can someone explain me this please I am a bit confused? It’s from a texbook I am reading (block cipher companion) it says that:

each weak key has $2^{32}$ fixed points $m$ where $operatorname{DES}_w(m) = m$ ( $w$ is a DES weak key). In other words encrypting a fixed point gives exactly the same point. To see this consider the $2^{32}$ ciphertexts after 8 rounds of encryption for which the two 32 bit halves are equal. For a weak DES key, the round keys of the eighth and ninth rounds are equal. Thus the intermediate texts after seven rounds and after nine rounds of encryption will be equal. Since the round keys of the 7th and the 10th rounds are equal, the intermediate texts after 6 rounds and after 10 rounds of encryption will be equal. Continuing like this it follows that the plaintext will be equal to the ciphertext.

Now did not quite get :

  1. Why are there $2^{32}$ fixed points? And why after the eighth round of encryption the two halves must be equal?
  2. What does the author mean by the intermediate texts exactly? And how are they being equal?

P.S.: This question is not asking for what the weak keys are. I am asking for “fixed points” not exactly weak keys. For this reason I think it is not a duplicate of any other question. There is one similar question to what I’ve asked for which I have checked and saw that no satisfactory answer has been given there either.
Here is the similar question with no proper answers:

What is the fixed point attribute of DES with weak keys?

Cryptography methods that allow for multiple keys to be used?

I’m currently working on a webapp in which I’d like all nearly all user data to be encrypted, but accessible by multiple users (ie a doc encrypted should be decrypt-able by parties A, B, C, all independently). Is there any way to achieve this? One issue I realized is that it would be nearly impossible::Listen

I’m currently working on a webapp in which I’d like all nearly all user data to be encrypted, but accessible by multiple users (ie a doc encrypted should be decrypt-able by parties A, B, C, all independently). Is there any way to achieve this?

One issue I realized is that it would be nearly impossible based on my knowledge to append users to have access to docs (ie allow party D to now access that prior document), as I’d need to re-encrypt the plaintext, but as I don’t have access to that, I couldn’t, correct?

How to find fixed points for DES weak keys

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