
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
Find the standard equation of the hyperbola that satisfies the following conditions:
1. $a=3$, $b=4$, and the foci are on the x-axis.
2. The coordinates of the foci are $(0,10)$ and $(0,-10)$, and the absolute difference in distances from any point on the hyperbola to the two foci is $16$.
3. The foci are $(0,-5)$ and $(0,5)$, and it passes through the point $\left( \dfrac {4 \sqrt {3}}{3},2 \sqrt {3}\right)$.

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