
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 $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with left and right foci $F_{1}$ and $F_{2}$, and a point $P(2,1)$ on the ellipse. If $|PF_{1}|+|PF_{2}|=2\sqrt{6}$, then $|F_{1}F_{2}|=\_\_\_\_\_\_$.

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