
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
If $\forall a$, $b$, $c \in A$, $f(a)$, $f(b)$, $f(c)$ are the lengths of the three sides of a triangle, then the function $f(x)$ on interval $A$ is called a "triangle function". Given that the function $f(x) = -\sin^2 x - \cos x + \frac{5}{4} + m$ is a "triangle function" on the interval $[-\frac{\pi}{3}, \frac{2\pi}{3}]$, please solve the following problems:  
$(1)$ The range of $f(x)$ on the interval $[-\frac{\pi}{3}, \frac{2\pi}{3}]$ is ______;  
$(2)$ The range of the real number $m$ is ______.

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