
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
Two skaters, Allie and Billie, are at points  $A$  and  $B$ , respectively, on a flat, frozen lake. The distance between  $A$  and  $B$  is  $100$  meters. Allie leaves  $A$  and skates at a speed of  $8$  meters per second on a straight line that makes a  $60^\circ$  angle with  $AB$ . At the same time Allie leaves  $A$ , Billie leaves  $B$  at a speed of  $7$  meters per second and follows the straight path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meeting Billie?

[asy]
defaultpen(linewidth(0.8));
draw((100,0)--origin--60*dir(60), EndArrow(5));
label(" $A$ ", origin, SW);
label(" $B$ ", (100,0), SE);
label(" $100$ ", (50,0), S);
label(" $60^\circ$ ", (15,0), N);[/asy]

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