
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

The hypotenuse of a right triangle is one side of a square located outside the triangle. Find the distance between the vertex of the right angle of the triangle and the center of the square if the legs of the triangle are $a$ and $b$.

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