
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
A function \( f \) satisfies the following conditions for all nonnegative integers \( x \) and \( y \):

- \( f(0, x) = f(x, 0) = x \)
- If \( x \geq y \geq 0 \), \( f(x, y) = f(x - y, y) + 1 \)
- If \( y \geq x \geq 0 \), \( f(x, y) = f(x, y - x) + 1 \)

Find the maximum value of \( f \) over \( 0 \leq x, y \leq 100 \).

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