
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, the vertex of angle $\alpha$ is at the origin, the initial side is the non-negative $x$-axis, and the terminal side passes through the point $(-1,2)$.  
$(1)$ Find the value of $\sin \alpha \cdot \tan \alpha$;  
$(2)$ Find the value of $\frac{sin(\alpha+\frac{\pi}{2}) \cdot cos(\frac{7\pi}{2}-\alpha) \cdot tan(2\pi-\alpha)}{sin(2\pi-\alpha) \cdot tan(-\alpha)}$.

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