
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 $p$: For all $x_1 \in \left(\frac{1}{2}, 2\right)$, there exists $x_2 \in \left(\frac{1}{2}, 2\right)$ such that the equation $\log _{2}x_{1}+a=x_{2}^{2}+2$ holds true. Proposition $q$: For all $x_{1}$, $x_{2} \in \left[0,1\right]$, the inequality $a+3x_2 > 4^{x_1}$ always holds. If proposition $p$ is true and proposition $q$ is false, then the range of real number $a$ is ______.

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