
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 parabola $n:y=\frac{1}{m}x^{2}-m\left(m\neq 0\right)$ intersects the $x$-axis from left to right at points $A$ and $B$, with the vertex at $C$.  
$(1)$ The shape of $\triangle ABC$ is ______.  
$(2)$ If $AB=4$, find the value of $m$.  
$(3)$ The line $y=\frac{2}{m}x$ intersects the parabola $n$ from left to right at points $D$ and $E$, where $P$ is the midpoint of segment $DE$, and $PF$ is parallel to the $y$-axis and intersects the parabola $n$ at point $F$.  
① If $PF=\frac{10}{3}$, find the value of $m$;  
② Find the area of $\triangle DEF$ (expressed as an algebraic expression involving $m$).

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