
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
$(1)$ If the inequality $m^{2}x^{2}-2mx \gt -x^{2}-x-1$ holds for all $x$, find the range of real numbers $m$. <br/>$(2)$ Solve the inequality $\left(x-1\right)\left(ax-1\right) \gt 0$, where $a \lt 1$.

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