Hiking and camping near Madrid

Hi all,
Over in Madrid beginning of September and my partner and I are renting a car for a day and wanting to go up to the mountains for a day hike, then camp for 1 night around Madrid.

We have done small day hikes in Sierra de Guadarrama before, however never camped - Im not sure you can camp?!?! So looking for some recommendations for a good 5/6 hr hike and somewhere to set up camp anywhere within an 1/2 hr drive from Madrid.
Thanks in advance,
Vicki

I just read that camping in the provence of Madrid is forbidden :( so amazing hike recommendations are welcome!!

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