
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 $a \gt 0$, $b \gt 0$, and $a+b=5ab$. Let proposition $p$ be: There exist $a \gt 0$, $b \gt 0$ such that $x\geqslant a+b$ holds true; proposition $q$: For any $a \gt 0$, $b \gt 0$, the inequality $xab\leqslant b^{2}+5a$ always holds.<br/>$(1)$ If proposition $p$ is true, find the range of real number $x$; <br/>$(2)$ If both proposition $p$ and proposition $q$ are true, find the range of real number $x$.

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