
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 proposition $p:\exists x\in \left(-1,1\right)$, $x^{2}-\left(a+2\right)x+2a=0$, proposition $q:x_{1}$ and $x_{2}$ are two real roots of the equation $x^{2}-2mx-3=0$, the inequality $a^{2}-3a\geqslant |x_{1}-x_{2}|$ holds for any real number $m\in \left[-1,1\right]$. <br/>$(1)$ If proposition $p$ is a true proposition, find the range of real number $a$; <br/>$(2)$ If propositions $p$ and $q$ have exactly one true proposition, find the range of real number $a$.

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