
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
Regarding the constant $\pi$, the history of mathematics has seen many creative methods for its estimation, such as the famous Buffon's Needle experiment and the Buffon-Laplace experiment. Inspired by these, we can also estimate the value of $\pi$ through the following experiment: Ask $200$ students to randomly write down a pair of positive real numbers $(x,y)$, both less than $1$, and then count the number of pairs $(x,y)$ that can form an obtuse triangle with $1$ as one of its sides. Let's denote this count as $m$; finally, estimate the value of $\pi$ based on the count $m$. If the result is $m=56$, then the estimated value of $\pi$ is $\boxed{\frac{78}{25}}$ (expressed as a fraction).

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