
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 proposition $p$: The solution set of the inequality $2^{x}+1 < a$ with respect to $x$ is $\varnothing$; proposition $q$: The domain of the function $y=\lg (ax^{2}-x+a)$ is $\mathbb{R}$.

$(1)$ If the proposition "$p \land q$" is true, find the range of the real number $a$;  
$(2)$ Let proposition $m$: The graph of the function $y=x^{2}+bx+a$ has common points with the $x$-axis. If $\neg p$ is a sufficient but not necessary condition for $\neg m$, find the range of the real number $b$.

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