
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
Function  $f(x, y): \mathbb N \times \mathbb N \to \mathbb Q$  satisfies the conditions:

(i)  $f(1, 1) =1$ ,

(ii)  $f(p + 1, q) + f(p, q + 1) = f(p, q)$  for all  $p, q \in  \mathbb N$ , and

(iii)  $qf(p + 1, q) = pf(p, q + 1)$  for all  $p, q \in  \mathbb N$ .

Find  $f(1990, 31).$ 

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