
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 circle in the first quadrant with center on the curve  $y=2x^2-27$  is tangent to the  $y$ -axis and the line  $4x=3y$ . The radius of the circle is  $\frac{m}{n}$  where  $M$  and  $n$  are relatively prime positive integers. Find  $m+n$ .

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