
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
Let  $q$  be a positive rational number. Two ants are initially at the same point  $X$  in the plane. In the  $n$ -th minute  $(n = 1,2,...)$  each of them chooses whether to walk due north, east, south or west and then walks the distance of  $q^n$  metres. After a whole number of minutes, they are at the same point in the plane (not necessarily  $X$ ), but have not taken exactly the same route within that time. Determine all possible values of  $q$ .

Proposed by Jeremy King, UK

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