
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  $q$  be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard:


- In the first line, Gugu writes down every number of the form  $a-b$ , where  $a$  and  $b$  are two (not necessarily distinct) numbers on his napkin.
- In the second line, Gugu writes down every number of the form  $qab$ , where  $a$  and  $b$  are
two (not necessarily distinct) numbers from the first line.
- In the third line, Gugu writes down every number of the form  $a^2+b^2-c^2-d^2$ , where  $a, b, c, d$  are four (not necessarily distinct) numbers from the first line.

Determine all values of  $q$  such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line.

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