
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 value of \\(\pi\\), the history of mathematics has seen many creative methods for its estimation, such as the famous Buffon's Needle experiment and the Charles' experiment. Inspired by these, we can also estimate the value of \\(\pi\\) through designing the following experiment: ask \\(200\\) students, each to randomly write down a pair of positive real numbers \\((x,y)\\) both less than \\(1\\); then count the number of pairs \\((x,y)\\) that can form an obtuse triangle with \\(1\\) as the third side, denoted as \\(m\\); finally, estimate the value of \\(\pi\\) based on the count \\(m\\). If the result is \\(m=56\\), then \\(\pi\\) can be estimated as \_\_\_\_\_\_ (expressed as a fraction).

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