
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
Three congruent isosceles triangles are constructed inside an equilateral triangle with a side length of $\sqrt{2}$. Each base of the isosceles triangle is placed on one side of the equilateral triangle. If the total area of the isosceles triangles equals $\frac{1}{2}$ the area of the equilateral triangle, find the length of one of the two congruent sides of one of the isosceles triangles.
A) $\frac{1}{4}$
B) $\frac{1}{3}$
C) $\frac{2\sqrt{2}}{5}$
D) $\frac{1}{2}$
E) $\frac{\sqrt{3}}{4}$

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