
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 $p$: the equation $x^{2}+y^{2}-4x+a^{2}=0$ represents a circle; $q$: the equation $\frac{{y}^{2}}{3}+\frac{{x}^{2}}{a}=1(a＞0)$ represents an ellipse with its focus on the $y$-axis.  
$(1)$ If $p$ is a true statement, find the range of real number $a$;  
$(2)$ If the statement $p\vee q$ is true and $p\wedge q$ is false, find the range of real number $a$.

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