
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 $f(x) = mx^{2} - (m+1)x + 1$.  
$(1)$ If $m > 0$, find the solution set of the inequality $f(x) < 0$.  
$(2)$ If $f(x) \leq 2$ holds for all $x \in [1,2]$, find the range of real numbers for $m$.

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