
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 right triangle with perpendicular sides  $a$  and  $b$  and hypotenuse  $c$  has the following properties: $a = p^m$  and  $b = q^n$  with  $p$  and  $q$  prime numbers and  $m$  and  $n$  positive integers,  $c = 2k +1$  with  $k$  a positive integer.
Determine all possible values of  $c$  and the associated values of  $a$  and  $b$ .

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