
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 quadratic function $y=ax^2-(a+1)x-4$ (where $a$ is a constant),  
(1) It is known that the vertex of the graph of the quadratic function $y=ax^2-(a+1)x-4$ is on the y-axis, find the value of $a$;  
(2) Upon investigation, it is found that regardless of the value of $a$, the graph of the quadratic function will always pass through two fixed points in the Cartesian coordinate system. Find the coordinates of these two points;  
(3) Given that one root of the quadratic equation $ax^2-(a+1)x-4=0$ is between -1 and 0 (excluding -1 and 0), and the other root is between 2 and 3 (excluding 2 and 3), find the integer value of $a$.

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