
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 $f(x)$ passes through the points $A(-1,0)$, $B(3,0)$, and $C(0,-3)$.
$(1)$ Find the analytical expression of $f(x)$;
$(2)$ When $x\in [-1,4]$, the inequality $f(x)-2m\leqslant 0$ always holds. Find the range of real numbers for $m$.

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