
Samsung Galaxy Note3
The latest firmware for Samsung Galaxy Note3 (SM-N9009) is N9009KEUDNC1 with Jelly(Android Bean), released on March 30, 2014 for 1 region (CSC). Every official firmware build for SM-N9009, 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/N9009KEUDNC1
SM-N9009 started on Android 4 and has received 1 major Android upgrade; it now runs Android 5.0 (Lollipop). The newest build, N9009KEUDNC1, was released on March 30, 2014. No new build has appeared for 153 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-N9009 runs Android 5.0 (Lollipop): build N9009KEUDNC1, released on March 30, 2014.
The latest build for SM-N9009, N9009KEUDNC1, was released on March 30, 2014 (12 years ago).
SM-N9009 has had no new firmware for 153 months, so regular updates have most likely ended.
Yes, all firmware files for SM-N9009 are 100% official, original, and obtained directly from Samsung servers.
To flash SM-N9009, 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-N9009.