
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 graph of the quadratic function $y=ax^{2}+bx+c$ passes through three points $A(1,5)$, $B(0,3)$, and $C(-1,-3)$.<br/>$(1)$ Find the analytical expression of this function;<br/>$(2)$ Use the completing the square method to find the coordinates of the vertex of this quadratic function's graph.

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