
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
Inside the triangle \( ABC \), a point \( M \) is taken such that the following property is known: if the sum of the squares of all sides of the triangle is added to three times the sum of all squares of the distances from point \( M \) to the vertices of the triangle, the resulting value does not exceed \( 24 \cdot x \). Find the side of the triangle \( y \), given that the area of the triangle \( ABC \) is at least \( \sqrt{3} \cdot x \). If necessary, round the found value to two decimal places.

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