
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
Mr. Jones teaches algebra.  He has a whiteboard with a pre-drawn coordinate grid that runs from  $-10$  to  $10$  in both the  $x$  and  $y$  coordinates.  Consequently, when he illustrates the graph of a quadratic, he likes to use a quadratic with the following properties:
I.  The quadratic sends integers to integers.
II.  The quadratic has distinct integer roots both between  $-10$  and  $10$ , inclusive.
III.  The vertex of the quadratic has integer  $x$  and  $y$  coordinates both between  $-10$  and  $10$ , inclusive.
How many quadratics are there with these properties?


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