
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 $m$, $x \in \mathbb{R}$, vectors $\overrightarrow{a} = (x, -m)$, $\overrightarrow{b} = ((m+1)x, x)$.  
$(1)$ When $m > 0$, if $|\overrightarrow{a}| < |\overrightarrow{b}|$, find the range of $x$;  
$(2)$ If $\overrightarrow{a} \cdot \overrightarrow{b} > 1 - m$ holds for any real number $x$, find the range of $m$.

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