
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 
$$q(x, y) = b_0 + b_1x + b_2y + b_3x^2 + b_4xy + b_5y^2 + b_6x^3 + b_7x^2y + b_8xy^2 + b_9y^3.$$ 
Suppose that 
$$q(0,0) = q(1,0) = q(-1,0) = q(0,1) = q(0,-1) = q(1,1) = q(-1,-1) = q(2,2) = 0,$$
and also 
$$\frac{\partial q}{\partial x}(1,1) = \frac{\partial q}{\partial y}(1,1) = 0.$$
Find a point $(u,v)$ such that $q(u,v) = 0$ for all such polynomials, where $u$ and $v$ are not integers.

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