
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 a line $l$ that passes through the point $M_{0}(x_{0},y_{0})$ with an angle of inclination $α$, and $M(x,y)$ as any point on line $l$, the parametric equation of line $l$ is $$\begin{cases} x=x_{0}+t\cos α, \\ y=y_{0}+t\sin α \end{cases}$$ where $t$ is a parameter.

(1) If $M_{1}$ and $M_{2}$ are two points on line $l$ with corresponding parameters $t_{1}$ and $t_{2}$, then $|{\overrightarrow{M_{0}M_{1}}}| |{\overrightarrow{M_{0}M_{2}}}|=|t_{1}t_{2}|$ and $|{\overrightarrow{M_{1}M_{2}}}|=|t_{2}-t_{1}|= \sqrt{(t_{2}+t_{1})^{2}-4t_{1}t_{2}}$.

(2) If the midpoint of segment $M_{1}M_{2}$ is $M_{3}$, and the corresponding parameters of points $M_{1}$, $M_{2}$, and $M_{3}$ are $t_{1}$, $t_{2}$, and $t_{3}$, respectively, then $t_{3}= \frac{t_{1}+t_{2}}{2}$.

(3) If the midpoint of segment $M_{1}M_{2}$ on line $l$ is $M_{0}(x_{0},y_{0})$, then $t_{1}+t_{2}=0$ and $t_{1}t_{2} < 0$.

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