
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 the parabola $y=ax^{2}+bx+3$ passes through point $A\left(3,0\right)$ and point $B\left(4,3\right)$.<br/>$(1)$ Find the relationship formula of this parabola;<br/>$(2)$ Write down its direction of opening, axis of symmetry, vertex coordinates, and maximum (or minimum) value.

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