
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 function $f(x)=-x^{3}+ax^{2}-4$, where $a$ is a real constant.<br/>$(1)$ When $a=3$, find the equation of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$;<br/>$(2)$ Discuss the monotonicity of $f(x)$;<br/>$(3)$ If there exists $x_{0}\in (0,+\infty)$ such that the inequality $f(x_{0}) > 0$ holds, find the range of real numbers for $a$.

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