Infer correct types when dynamically accessing values in nested object

I want to create a function that you can give a list of object keys and it would navigate through those keys on the object you pass and return the value or undefined if it doesn't exist and infer the correct type based on the type of the object passed in.

Here's how it should work:

prop(['one', 'two', 'three'], {one: {two: {three: 'found it!'}}}) === 'found it!'
prop(['one', 'two', 'four'], {one: {two: {three: 'found it!'}}}) === undefined

I've had a go at typing it specifically with 3 levels here but want it to work with any reasonable depth in the end. Here's where I got to so far, see it with TS errors here :

function prop<T, K1 extends keyof T, K2 extends keyof T[K1], K3 extends keyof T[K1][K2]>(keys: [K1, K2, K3], obj: T) {
  // I don't think this type is right but can't find a way to get it to understand what level it's currently at
  let currentDepth: T | T[K1] | T[K1][K2] = obj;
  for (let i = 0; i < keys.length; i++) {
    const k = keys[i]
    currentDepth = currentDepth[k];
    if (i === keys.length - 1) {
      return currentDepth;
    }
    if (typeof currentDepth === 'undefined') {
      return currentDepth
    }
  }
  return undefined
}

Any hints about whether is possible or other approaches appreciated. Thanks