
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
Draw the perpendicular to the tangent line of the ellipse  $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\ (a>0,\ b>0)$  from the origin  $O(0,\ 0).$ 

Let  $\theta$  be the angle between the perpendicular and the positive direction of  $x$  axis. Denote the length of the perpendicular by  $r(\theta).$ 

Calculate  $\int_{0}^{2\pi}r(\theta )^{2}\ d\theta.$ 

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