
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]


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Regarding the mathematical constant \(\pi\), there have been many creative methods for estimating its value throughout history, such as the Buffon's needle experiment and the Monte Carlo method. Inspired by these, we can design an experiment to estimate the value of \(\pi\): First, ask 200 students to each randomly write down a pair of positive real numbers \((x, y)\) that are both less than 1. Then, count the number of pairs \((x, y)\) that can form an obtuse triangle with 1 as one of its sides, denoting this count as \(m\). Finally, use the count \(m\) to estimate the value of \(\pi\). If the result of the count is \(m=56\), then we can estimate \(\pi \approx\) \_\_\_\_\_\_\_\_\_. (Express your answer as a fraction.)

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