
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 that the parabola $y=ax^{2}-bx+3$ passes through points $A(1,2)$ and $B(2,3)$.  
$(1)$ Find the analytical expression of this parabola.  
$(2)$ Determine whether point $C(-1,5)$ lies on this parabola.

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