
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 an ellipse $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{96}=1(a＞4\sqrt{6})$ with the left and right foci $F_{1}$ and $F_{2}$ respectively. Let $P$ be a point on the ellipse $C$ such that $|PF_{1}|:|PF_{2}|:|OF_{2}|=8:6:5$ (where $O$ is the origin). Find $|PF_{1}|-|PF_{2}|$.

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