
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 $f(x) = |x+1| + |mx-1|$.  
(1) If $m=1$, find the minimum value of $f(x)$, and specify the range of $x$ for which this occurs;  
(2) If $f(x) \geq 2x$, find the range of values for $m$.

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