
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 each positive integer  $m$  and  $n$  define function  $f(m, n)$  by  $f(1, 1) = 1$ ,  $f(m+ 1, n) = f(m, n) +m$  and  $f(m, n + 1) = f(m, n) - n$ . Find the sum of all the values of  $p$  such that  $f(p, q) = 2004$  for some  $q$ .

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