Decompose the Toffoli gate

Build a three-qubit circuit whose unitary equals $\mathrm{CCX}$ (controls qubits $0$ and $1$, target qubit $2$) up to a global phase, using only $H$, $T=\mathrm{diag}(1,e^{i\pi/4})$, $T^\dagger=\mathrm{diag}(1,e^{-i\pi/4})$ and $\mathrm{CNOT}$, with at most $6$ CNOTs and at most $15$ gates, and no measurements. The grader computes the $8\times8$ unitary $U$ of your circuit and the process fidelity $F_U=\lvert\mathrm{Tr}(\mathrm{CCX}^\dagger U)\rvert/8$, which equals $1$ exactly when $U=e^{i\gamma}\,\mathrm{CCX}$. Qubit $0$ is the most significant bit, so $\lvert a,b,c\rangle$ has index $4a+2b+c$.