
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  $X=\{2^m3^n|0 \le m, \ n \le 9 \}$ . How many quadratics are there of the form  $ax^2+2bx+c$ , with equal roots, and such that  $a,b,c$  are distinct elements of  $X$ ?

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