Solution:

Let's assume the sum of the original sequence to be S.

Given, average = 300

Therefore, S/N = 300

Hence, S = 300N

After replacing a1 with 6a1, the new sequence becomes {6a1, a2, a3, ..., aN}

Let's assume the sum of the new sequence to be S'.

Given, new average = 400

Therefore, S'/N = 400

Hence, S' = 400N

We know that the sequence is non-decreasing.

Therefore, a1 ≤ a2 ≤ a3 ≤ ... ≤ aN

Case 1: a1 < a2="" />< a3="" />< ...="" />< />

In this case, we can say that the original sequence and the new sequence are the same.

Hence, a1 = 6a1, which is not possible.

Case 2: a1 = a2 = a3 = ... = aN

In this case, we can say that the original sequence is {a1, a1, a1, ..., a1} and the new sequence is {6a1, a1, a1, ..., a1}

From the given data, we can write:

(6a1 + (N-1)a1)/N = 400

Simplifying, we get:

7a1 = 400

a1 = 57.14

But a1 has to be a positive integer, which is not possible.

Case 3: a1 ≤ a2 ≤ a3 ≤ ... ≤ aN-1 < />

In this case, we can say that the original sequence is {a1, a2, a3, ..., aN-1, aN-1} and the new sequence is {6a1, a2, a3, ..., aN-1, aN-1}

From the given data, we can write:

(6a1 + (N-2)a1 + 2aN-1)/N = 400

Simplifying, we get:

7a1 + 2aN-1 = 800

aN-1 = (800 - 7a1)/2

Since aN-1 is a positive integer, (800 - 7a1) has to be an even multiple of 2.

Therefore, 800 - 7a1 = 2, 4, 6, ..., 798, 800

Solving for a1, we get:

a1 = 114, 113, 112, ..., 2, 1

Therefore, the number of possible values of a1 is 114 - 1 + 1 = 114.

But we need to check if all these possible values of a1 satisfy the non-decreasing sequence condition.

For example, if a1 = 114, then the original sequence is {114, 114, 114, ..., 114} and the new sequence is {684, 114, 114, ..., 114}

In this case, the sequence is non-decreasing.

Similarly, we can check for all the possible values of a1.

Hence, the number of possible values of a1 is 1.