
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 points $F_1(-3,0)$ and $F_2(3,0)$. If the locus of point $M$ satisfying the condition $|MF_1|+|MF_2|=2m+1$ is an ellipse, and the locus of point $N$ satisfying the condition $\left| |NF_1|-|NF_2| \right| = 2m-1$ is a hyperbola, then the range of the real number $m$ is __________.

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