
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 parabola about $x$, $y=-\frac{1}{2}x^{2}+bx+c$ passes through the points $\left(1,0\right)$ and $\left(0,\frac{3}{2}\)$.  
$(1)$ Find the analytical expression of the parabola.  
$(2)$ Shift the parabola so that its vertex coincides with the origin. Provide a method for the shift and the analytical expression of the parabola after the shift.

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