
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
Exponential growth, also known as explosive growth, has a conclusion that when $a \gt 1$, the average rate of change of the exponential function $y=a^{x}$ on the interval $\left[0,t\right]$ increases as $t$ increases. Given real numbers $a$ and $b$ that satisfy $a^{b}b^{a}+\log _{a}b=0$.

$(1)$ Compare the sizes of $ab+1$ and $a+b$;

$(2)$ When $a=2$, compare the sizes of $2^{b}$ and $\frac{1}{b}$;

$(3)$ When $a=2$, determine the sign of $({2b+1-\sqrt{5}})({3b-2})$.

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