
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
How many ordered pairs  $(x, y)$  of real numbers  $x$  and  $y$  are there such that  $-100 \pi \le x \le 100 \pi$ ,  $-100 \pi \le y \le 100 \pi$ ,  $x + y = 20.19$ , and  $\tan x + \tan y = 20.19$ ?


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