The inside lane of a track has length 400 m and the outside lane has length less than 500 m. From the marked line, as shown in the diagram

Let \( L \) be the length of the outside lane in meters. According to the problem:

- The inside lane has a length of 400 m.
- Lynn runs at a speed that is 3 times that of Max.

Given that both start from the same point at the marked line, they run counterclockwise until they first meet back at that point after Max has completed 3 laps.

**Speed and Distance Analysis:**

1. Let \( v \) be the speed of Max.
2. Then, Lynn's speed is \( 3v \).

**Time Taken for Max to Complete 3 Laps:**

The time taken by Max to complete 3 laps is:
\[
\text{Time}_{\text{Max}} = \frac{\text{Distance}}{\text{Speed}} = \frac{3 \times 400}{v} = \frac{1200}{v} \text{ seconds}
\]

**Distance Covered by Lynn in the Same Time:**

In the same time, Lynn runs:
\[
\text{Distance}_{\text{Lynn}} = \text{Speed} \times \text{Time} = 3v \cdot \frac{1200}{v} = 3600 \text{ meters}
\]

**Finding the Number of Laps for Lynn:**

The number of laps Lynn completes is given by:
\[
\text{Laps}_{\text{Lynn}} = \frac{\text{Distance}_{\text{Lynn}}}{L} = \frac{3600}{L}
\]

**Condition for Meeting Back at the Marked Line:**

For both to meet back at the marked line at the same time, Lynn’s total distance must be an integer number of laps plus an additional lap full cycle of the track, which means:
\[
3600 \equiv 0 \, (\text{mod} \, L)
\]

and we also know that after some integer number of laps, Max has completed exactly 3 laps.

Since Lynn runs 3600 m in the same time Max runs 1200 m, we can establish the relationship:
\[
\frac{3600}{L} = 3 + n \quad \text{for some integer } n
\]

This implies that:
\[
3600 = L(3 + n)
\]
(or)
\[
L = \frac{3600}{3 + n}
\]

Considering \( L < 500 \):
\[
\frac{3600}{3+n} < 500
\]
\[
3600 < 500(3+n)
\]
\[
\frac{3600}{500} < 3+n
\]
\[
7.2 < 3+n \quad \Rightarrow \quad n > 4.2
\]

Thus, \( n \) must be at least 5. Testing integer values leads us to \( n = 5 \):
\[
L = \frac{3600}{3 + 5} = \frac{3600}{8} = 450 \text{ meters}
\]

Finally, validate if any other values can yield lengths under 500:
- For \( n = 6 \): \( L = \frac{3600}{9} = 400 \) (not valid as the inside lane length).
- For \( n = 7 \): \( L < 500 \), but gives invalid lengths until \( n = 8 \).

Hence, the only valid length of the outside lane, rounded to the nearest meter, is:
\[
\boxed{450} \text{ meters}
\]