
Samsung GALAXY Note 3 Neo LTE
The latest firmware for Samsung GALAXY Note 3 Neo LTE (SM-N7509V) is N7509VKEUAOA2 with Android 4.3 (Jelly Bean), released on January 21, 2015 for 1 region (CSC). Every official firmware build for SM-N7509V, 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/N7509VKEUAOA2
Specifications of SM-N7505, a regional variant of the same device. Chipset and SIM lines are left out: they can differ by region.

SM-N750S

SM-N950UZK

SM-N770X

SM-N960W

SM-N930V
You can find the firmware on the model: SM-N7509V page.
Yes, all firmware files for SM-N7509V are 100% official, original, and obtained directly from Samsung servers.
To flash SM-N7509V, 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-N7509V.