
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
Let  $ABC$  be an equilateral triangle with side length  $10$ . A square  $PQRS$  is inscribed in it, with  $P$  on  $AB, Q, R$  on  $BC$  and  $S$  on  $AC$ . If the area of the square  $PQRS$  is  $m +n\sqrt{k}$  where  $m, n$  are integers and  $k$  is a prime number then determine the value of  $\sqrt{\frac{m+n}{k^2}}$ .

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