
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
In a new race setup, all runners start at point \( A \), touch any part of the 800-meter wall, and stop at point \( B \). The distance from \( A \) directly down to the left end of the wall is 200 meters. The distance from the right end of the wall directly vertically up to \( B \) is 400 meters. Determine the number of meters in the minimum distance a participant must run, rounding to the nearest meter. Illustrate using a diagram similar to the original problem's setup:

[asy]
import olympiad; import geometry; size(250);
defaultpen(linewidth(0.8));
draw((0,2)--origin--(8,0)--(8,4));
label("200 m",(0,2)--origin,W); label("800 m",(0,0)--(8,0),S); label("400 m",(8,0)--(8,4),E);
draw((0,2)--(4,0)--(8,4),linetype("3 3")+linewidth(0.7));
label("$A$",(0,2),N); label("$B$",(8,4),N);
[/asy]

Present the answer in LaTex format: \boxed{Your answer}
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