Find the least positive value of $\theta$.
\[
\cot \left(6 \theta+6^{\circ}\right)=\frac{1}{\tan \left(7 \theta+3^{\circ}\right)}
\]
\[
\theta=
\]

Step 1 ：The cotangent of an angle is the reciprocal of the tangent of that angle. Therefore, the equation can be simplified to: \(\tan \left(6 \theta+6^{\circ}\right)=\tan \left(7 \theta+3^{\circ}\right)\)

Step 2 ：The tangent function has a period of 180 degrees, so the two angles are equivalent if they differ by a multiple of 180 degrees. Therefore, we can write: \(6 \theta+6^{\circ} = 7 \theta+3^{\circ} + k \cdot 180^{\circ}\) where k is an integer.

Step 3 ：Solving this equation for \(\theta\) gives us \(\theta = 3 - 180k\). However, we want the least positive value of \(\theta\), so we need to find the smallest positive integer value of k that makes \(\theta\) positive.

Step 4 ：The solution to the equation is \(\theta = 3 - 180k\). However, we want the least positive value of \(\theta\), so we need to find the smallest positive integer value of k that makes \(\theta\) positive.

Step 5 ：The least positive value of \(\theta\) is \(\boxed{3^{\circ}}\)