elementary number theory - A simple modular arithmetic query.

Given $a,b,c\in\Bbb N$ with $\mathsf{gcd}(a,b)=\mathsf{gcd}(b,c)=\mathsf{gcd}(c,a)=1$ we know that there are $m_1,m_2,m_3\in\Bbb N$ such that $a\equiv m_1a^2\bmod abc$, $ab\equiv m_2ab\bmod abc$ and $b\equiv m_3b^2\bmod abc$ holds.

It is also easy to see there is a single $m$ such that

$$a\equiv ma\bmod ab,\quad b\equiv mb\bmod ab$$ holds.

However how to find a single $m$ coprime to $c$ such that

$$1\equiv ma\bmod abc,\quad 1\equiv mb\bmod abc$$ holds?

At least how to find a single $m$ such that

$$\ell_1 a\equiv ma^2\bmod abc, \quad \ell_2 ab\equiv mab\bmod abc,\quad\ell_3b\equiv mb^2\bmod abc$$ holds where $0<\ell_1,\ell_2,\ell_3<\log abc$ holds and at least one of $\ell_1,\ell_2,\ell_3$ is distinct?

If not how small can we make $\max(\ell_1,\ell_2,\ell_3)$ where $a\nmid\ell_1$ and $b\nmid\ell_3$ holds?