
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 that $(x_{0}$,$y_{0})$ is a point on the circle $x^{2}+y^{2}=r^{2}$, then the line ${x_0}x+{y_0}y={r^2}$ is tangent to the circle $x^{2}+y^{2}=r^{2}$, and $(x_{0}$,$y_{0})$ is the point of tangency. Similarly, if the point $(x_{0}$,$y_{0})$ is a point on the ellipse $\frac{{{x^2}}}{{{a^2}}}+\frac{{{y^2}}}{{{b^2}}}=1({a>b>0})$, find the equation of the tangent line tangent to the ellipse at $(x_{0}$,$y_{0})$.

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