
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 the function $f(x)=\sqrt{\frac{{1-x}}{{3x+1}}}$ with domain set $A$, and the function $g(x)=\sqrt{{x^2}-(2a+1)x+{a^2}+a}$ with domain set $B$, <br/>$(1)$ when $a=0$, find $A\cap B$; <br/>$(2)$ Let proposition $p:x\in A$, proposition $q:x\in B$, $p$ is a sufficient but not necessary condition for $q$, find the range of real number $a$.

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