
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 the function $f(x)=|x-1|+|2x-1|$.
$(1)$ Solve the inequality $f(x) > 3-4x$;
$(2)$ If $f(x)+|1-x| \geqslant 6m^{2}-5m$ holds for all real numbers $x$, find the range of values for $m$.

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