
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 a point on the hyperbola $\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1$ ($a > 0,b > 0$) such that the absolute difference of the distances from the point to the left and right foci of the hyperbola is $4$, and if two points $A(x_{1},y_{1})$, $B(x_{2},y_{2})$ on the parabola $y=ax^{2}$ are symmetric about the line $y=x+m$, and $x_{1}x_{2}=-\dfrac{1}{2}$, then the value of $m$ is \_\_\_\_.

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