
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

Find a number that, when added to 13600, results in a perfect square. To solve this geometrically, consider a rectangle with an area of 13600 with one side equal to 136. Divide it into a square \( A \) and a rectangle \( B \). Cut the rectangle \( B \) into two equal rectangles, both denoted by \( C \). Position the rectangles \( C \) on two consecutive sides of the square \( A \).

(a) What is the area of rectangle \( C \)?

(b) Notice that if we add square \( X \), we complete a larger square. What should be the side length of square \( X \)?

(c) After answering the two previous questions, determine a number that, when added to 13600, results in a perfect square and find the square root of this perfect square.

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