
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 which non-negative integers  $ a<2007$  the congruence	 $ x^2\plus{}a \equiv 0 \mod 2007$  has got exactly two different non-negative integer solutions? 
That means, that there exist exactly two different non-negative integers  $ u$  and  $ v$  less than  $ 2007$ , such that  $ u^2\plus{}a$  and  $ v^2\plus{}a$  are both divisible by  $ 2007$ .

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