
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
Let  $S=\{p/q| q\leq 2009, p/q <1257/2009, p,q \in \mathbb{N} \}$ . If the maximum element of  $S$  is  $p_0/q_0$  in reduced form, find  $p_0+q_0$ .

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