
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
The quadratic function is $y=-\frac{1}{2}{x}^{2}-x-\frac{5}{2}$.  
$(1)$ Factorize the quadratic function to vertex form;  
$(2)$ The vertex of the quadratic function is ______;  
$(3)$ When $x \gt -1$, $y$ increases as $x$ ______;  
$(4)$ When $-3\leqslant x\leqslant 2$, the range of $y$ is ______.

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