
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 two propositions, $P$: "The equation $y^{2}=(a^{2}-4a)x$ represents a parabola with the focus on the negative half of the $x$-axis." Proposition $Q$: "The equation $x^{2}-x+a=0$ has real roots for $x$."  
$(1)$ If proposition $P$ is true, find the range of real numbers for $a$.  
$(2)$ If $P\vee Q$ is true and $P\wedge Q$ is false, find the range of real numbers for $a$.

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