
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
For positive reals \( p \) and \( q \), define the remainder when \( p \) is divided by \( q \) as the smallest nonnegative real \( r \) such that \( \frac{p - r}{q} \) is an integer. For an ordered pair \( (a, b) \) of positive integers, let \( r_1 \) and \( r_2 \) be the remainder when \( a \sqrt{2} + b \sqrt{3} \) is divided by \( \sqrt{2} \) and \( \sqrt{3} \) respectively. Find the number of pairs \( (a, b) \) such that \( a, b \leq 20 \) and \( r_1 + r_2 = \sqrt{2} \).

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