
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 three-dimensional Cartesian coordinate system $O-xyz$, given a point $P(x_{0}, y_{0}, z_{0})$ and a plane $\alpha$ with a normal vector $\overrightarrow{n}=(a, b, c)$, the equation of the plane is $a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0$. Solve the following problem using the above information: Given the equation of the plane $\alpha$ as $x+2y-2z+1=0$, the line $l$ is the intersection of the two planes $x-y+3=0$ and $x-2z-1=0$. Write down a direction vector of the line $l$ as ______, and the sine value of the angle between the line $l$ and the plane $\alpha$ is ______.

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