
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 standard equation of an ellipse $\frac{{x}^{2}}{5}+\frac{{y}^{2}}{4}=1$, let $({F}_{1},{F}_{2})$ denote the left and right foci of the ellipse, $O$ the origin, and $P$ a point on the ellipse in the first quadrant. Determine the range of values for $|P{F}_{1}|-|P{F}_{2}|$.

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