Muhammad,
The problem is that there are several 'Lenovo Chromebook 500e' devices using at least 4 different boards and AUE dates, see below:
> name=Lenovo 500e Chromebook
> hwidmatch=^ROBO360 .*
> file=chromeos_15886.44.0_coral_recovery_stable-channel_mp-v23.bin
> name=Lenovo 500e Chromebook Gen 3
> hwidmatch=^BOTEN-YGHA.*
> file=chromeos_16433.41.0_dedede_recovery_stable-channel_DededeMPKeys-v57.bin
> name=Lenovo 500e Chromebook Gen 4s
> hwidmatch=^PUJJOGA-TZXN.*
> file=chromeos_16433.41.0_nissa_recovery_stable-channel_NissaMPKeys-v66.bin
> name=Lenovo 500e Chromebook Gen 4s (Intel-TWL-N)
> hwidmatch=^PUJJOGATWIN-TKNY.*
> file=chromeos_16433.41.0_nissa_recovery_stable-channel_NissaMPKeys-v66.bin
> name=Lenovo 500e Yoga Chromebook Gen 4
> hwidmatch=^PUJJO-KTLR.*
> file=chromeos_16433.41.0_nissa_recovery_stable-channel_NissaMPKeys-v66.bin
> name=Lenovo 300e/500e Chromebook 2nd Gen
> hwidmatch=^PHASER360 [ABCD][A-Z0-9].*|^PHASER360 [G|H|K|L|M].{11}B.*
> file=chromeos_16433.41.0_octopus_recovery_stable-channel_OctopusMPKeys-v37.bin
> name=Lenovo Chromebook C340-11/300e/500e Chromebook 2nd Gen
> hwidmatch=^PHASER360 [E|F][A-Z0-9].*|^PHASER360 [K|L|M|N].{11}C.*
> file=chromeos_16433.41.0_octopus_recovery_stable-channel_OctopusMPKeys-v37.bin
If you could give us the HWID / codename it would help narrow it down.
However, if the device was previously opted-in to extended updates, there may not be any way to reverse it now.
The way I understand it, there was a 'firmware bump' that prevents reenabling Google Play Services once they've been opted-in on the device.
Once you know the board / codename you could try using an older recovery image, one prior to the extended update offer, and see if you can reenable Google Play Services then. You can find historical recovery images for your device on the site linked below:
~DennyL