
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
Let $p:\forall x\in R$, $ax^{2}+ax+1 \gt 0$, $q:\exists x\in \left[1,+\infty \right)$, $4^{x}+2^{x+1}-7-a \lt 0$.  
$(1)$ If $p$ is a true proposition, find the range of real number $a$;  
$(2)$ If $p\vee q$ is a true proposition and $q\wedge q$ is a false proposition, find the range of real number $a$.

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