
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 $p: \forall x\in R$, $ax^{2}+2ax+3 \gt 0$ holds true, and proposition $q: \exists x\in R$, $x^{2}+2ax+a+2=0$. If both propositions $p$ and $q$ are true, then the range of real number $a$ is ________. (Express the answer in interval notation)

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