where is the binary for fastboot mode located and How can I get a copy of it?

Question

Edit

Thanks to everyone in the commen section, lk.img seems to be what I was looking for and it is indeed right, because it has strings for varies commands like

oem p2u
oem reboot-recovery
oem off-mode-charge

the lk.img for my phone can be downloaded here

The only thing left is to reverse engineer it but most programs I use cannot figure out what file type it is

Motive

I am trying to find the binary for fastboot mode in my phone So that I can understand what my phone 's vendor fastboot does when I send a command using fastboot protocol

what Have I Tried

1

I read fastboot's source code here

in the Android.bp

cc_binary {
    name: "fastbootd",
    defaults: ["fastboot_defaults"],

    recovery: true,

    product_variables: {
        debuggable: {
            cppflags: ["-DFB_ENABLE_FETCH"],
        },
    },

    srcs: [
        "device/commands.cpp",
        "device/fastboot_device.cpp",
        "device/flashing.cpp",
        "device/main.cpp",
        "device/usb.cpp",
        "device/usb_client.cpp",
        "device/tcp_client.cpp",
        "device/utility.cpp",
        "device/variables.cpp",
        "socket.cpp",
    ],

it seems that the fastboot's program on the android phone will be compiled with name fastbootd,

So I tried finding it by downloading my phone's official firmware and locating it using linux's find but it doesn't seem to reside in the firmware

2

I have read similar question

But it didn't tell in more detail on how to get a copy of the binary and doesn't seem to have satisfactory answer

Phone details

Not sure if it matters because I think all android's fastboot binary should be in the same location

Model Number : vivo1819
vivo ROM : Funtouch OS 10.5
Android version : 11
Hardware Version No. : MP_0.1
Software Version : PD1831F_EX_A_8.1.48

Firmware Download

https://drive.google.com/file/d/13coKM6yMS39MVRauyyZ7FO3B_twlTB1a/view?usp=sharing

Question

So how can I get my phone's fastboot so that I can get a copy and learn more about it via reverse engineering