
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
Given that the line $AB$ is a common tangent to the curve $y=-\frac{1}{x}$ and the parabola $y^{2}=2px\left(p \gt 0\right)$, with tangency points $A(x_{1}$,$y_{1})$ and $B(x_{2}$,$y_{2})(x_{2} \gt 0)$, then $x_{1}y_{1}=$______, if $|AB|=\frac{{3\sqrt{10}}}{2}$, then $p=\_\_\_\_\_\_$.

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