
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
In the Cartesian coordinate system $xOy$, a moving line $l$: $y=x+m$ intersects the parabola $C$: $x^2=2py$ ($p>0$) at points $A$ and $B$, and $\overrightarrow {OA}\cdot \overrightarrow {OB}=m^{2}-2m$.

1. Find the equation of the parabola $C$.
2. Let $P$ be the point where the line $y=x$ intersects $C$ (and $P$ is different from the origin), and let $D$ be the intersection of the tangent line to $C$ at $P$ and the line $l$. Define $t= \frac {|PD|^{2}}{|DA|\cdot |DB|}$. Is $t$ a constant value? If so, compute its value; otherwise, explain why it's not constant.

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