
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
Suppose that \( X \) and \( Y \) are angles with \( \tan X = \frac{1}{m} \) and \( \tan Y = \frac{a}{n} \) for some positive integers \( a, m, \) and \( n \). Determine the number of positive integers \( a \leq 50 \) for which there are exactly 6 pairs of positive integers \( (m, n) \) with \( X + Y = 45^{\circ} \).

(Note: The formula \( \tan (X + Y) = \frac{\tan X + \tan Y}{1 - \tan X \tan Y} \) may be useful.)

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