Proving that a recursive sequence is bounded and monotone

Let $a_0=2\sqrt3$ and $b_0=3$, and define two sequences recursively by $$a_n=\frac{2a_{n-1}b_{n-1}}{a_{n-1}+b_{n-1}}\quad\text{and}\quad b_n=\sqrt{a_nb_{n-1}}.$$

This is an exercise in Jay Cummings (Real Analysis: A Long-Form Mathematics Textbook).The problem presents two separate recursive sequences defined in terms of each other. We are meant to show that one is monotonically increasing, one is monotonically decreasing, prove they both converge and then show that the value they converge to is $\pi$.

I have tried induction, I have tried to subtract successive terms, I have tried to first show they are bounded, to no avail.

A hint or the solution would be appreciated!