
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
**Background:** A new operation "$\odot$" is defined for any rational numbers $m$ and $n$ as follows: $m\odot n=mn(m-n)$. For example, $1\odot 2=1\times 2\times (1-2)=-2$. 

**Generalization:** 
(1) Simplify first, then find the value of $\left(a+b\right)\odot \left(a-b\right)$, where $a=\frac{1}{2}$ and $b=-1$. 
(2) If $x^{2}y\odot \left(x\odot y\right)=x^{p}y^{q}-x^{q}y^{p}$, find the values of $p$ and $q.

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