
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
A frog trainer places one frog at each vertex of an equilateral triangle  $ABC$  of unit sidelength. The trainer can make one frog jump over another along the line joining the two, so that the total length of the jump is an even multiple of the distance between the two frogs just before the jump. Let  $M$  and  $N$  be two points on the rays  $AB$  and  $AC$ , respectively, emanating from  $A$ , such that  $AM = AN = \ell$ , where  $\ell$  is a positive integer. After a finite number of jumps, the three frogs all lie in the triangle  $AMN$  (inside or on the boundary), and no more jumps are performed.
Determine the number of final positions the three frogs may reach in the triangle  $AMN$ . (During the process, the frogs may leave the triangle  $AMN$ , only their nal positions are to be in that triangle.)

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