
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
The transformation between finite and infinite is an important method in mathematics, as seen in the "Circle Field Method" in the "Nine Chapters on the Mathematical Art" (commentary by Liu Hui): "If it is cut again and again until it cannot be cut, then it merges with the circle completely without any loss." This illustrates that the "method of cutting a circle" is a process of transforming between the infinite and the finite. Similarly, in $\sqrt{2+ \sqrt{2+ \sqrt{2+\cdots}}}$, the "$\cdots$" represents an infinite repetition, but the original expression has a fixed value $x$, which can be determined by the equation $2 + x = x^2$ to be $x=2$. Similarly, converting the repeating decimal $0.\overset{\cdot}{3}\overset{\cdot}{6}$ into a fraction results in ______.

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