
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  $m$  and  $n$  be arbitrary positive integers, and 
  $a, b, c$  be different natural numbers of the form  $2^m.5^n$ .  Determine
 the number of all equations of the form  $ax^2-2bx+c=0$  if it is known that each equation
 has only one real solution.

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