
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
\begin{align*}
q(0,0) &=q(1,0) = q(-1,0) = q(0,1) = q(0,-1)= q(1,1) = q(1,-1) = q(3,3) = 0.
\end{align*}There is a point $(u,v)$ for which $q(u,v) = 0$ for all such polynomials, where $u$ and $v$ are not integers. Find the point $(u,v)$.

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