
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 proposition $p:∀x∈\{x|-2＜x≤3,x∈Z\},\frac{1}{3}{x}^{2}＜2a-3$, proposition $q:\exists x\in R$,$x^{2}+2x+a=0$.<br/>$(1)$ Write down the negation of proposition $p$; if proposition $q$ is false, find the range of real number $a$; <br/>$(2)$ Does there exist a real number $a$ such that propositions $p$ and $q$ have exactly one true proposition? If yes, find the range of real number $a$; if not, please explain the reason.

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