
Samsung GALAXY Note 3
The latest firmware for Samsung GALAXY Note 3 (SM-N9008S) is N9008SZCUCPA1 with Android 5.0 (KitKat), released on January 18, 2016 for 1 region (CSC). Every official firmware build for SM-N9008S, newest first: one row per build with all regions (CSC) that received it.
Each firmware build has its own page at https://samfirm.shop/build/{PDA}, where {PDA} is the build number from the list above. Example: https://samfirm.shop/build/N9008SZCUCPA1
SM-N9008S started on Android 4 and has received 1 major Android upgrade; it now runs Android 5.0 (KitKat). The newest build, N9008SZCUCPA1, was released on January 18, 2016. No new build has appeared for 131 months, so regular updates for this model have most likely ended.
Specifications of SM-N900, a regional variant of the same device. Chipset and SIM lines are left out: they can differ by region.

SM-N900V

SM-N910V

SM-N915V

SM-N916S

SM-N920AZW
The newest firmware for SM-N9008S runs Android 5.0 (KitKat): build N9008SZCUCPA1, released on January 18, 2016.
The latest build for SM-N9008S, N9008SZCUCPA1, was released on January 18, 2016 (10 years ago).
SM-N9008S has had no new firmware for 131 months, so regular updates have most likely ended.
Yes, all firmware files for SM-N9008S are 100% official, original, and obtained directly from Samsung servers.
To flash SM-N9008S, you need to download the appropriate firmware, put your device into Download Mode, and use the Odin tool to load the BL, AP, CP, and CSC files.
Downgrading is only possible if the bootloader version (binary bit) of the old firmware matches the current one. Always check the binary version before flashing SM-N9008S.