According to Thurston's construction, which can be found for instance in Farb-Margalit's *A Primer on Mapping Class Groups*, theorem 14.1 (here is a link to the version I am using: http://www.maths.ed.ac.uk/~aar/papers/farbmarg.pdf), If there are two curves $a_1$ and $a_2$ on a surface $S_{g,n}$ (a surface of genus $g$ with $n$ boundary components) that fill the surface, then we will be able to figure out the type of the element (periodic, reducible, Anosov) $\phi\in MCG(S_{g,n},\partial S_{g,n})$, $\phi=t_{a_2}t_{a_1}$, based on the intersection number of $a_1$ and $a_2$.

I want to know if anything can be said about the type of an element in the mapping class group that consists of three Dehn twists, say $\psi=t_{a_3} t_{a_2} t_{a_1}$, that $a_1$ and $a_2$ fill the surface and $t_{a_3}\notin <t_{a_1},t_{a_2}>$. Is there any general way we can determine whether such an element $\psi$ is periodic, reducible or Anosov (perhaps based on mutual intersections of these three curves)?

PS: I am actually working on a concrete example, so that $\psi=t_bt_at_c$, $\psi\in MCG(D_4,\partial D_4)$ ($D_4$ representing a $4$-puncutred disc) $a$ and $c$ fill the surface and $t_at_c$ is pseudo-Anosov, but $t_b\notin <t_a,t_c>$ as in the following figure: