Polynomial transformation on block encodings (Hermitian)

Let $P = \sum_{j=0}^{L} c_j x^j$ be a degree-$L$ complex polynomial such that $|P(x)| \leq 1$ for all $x \in [-1,1]$. Suppose $U$ is a $(1, m, 0)$-block-encoding of an $n$-qubit Hermitian matrix $A$. There exists a quantum circuit $\mathcal{V}$ that implements a $(4, m', 0)$-block-encoding of $P(A)$ for $m \leq m' \leq m + 3$, where $P(A)=\sum_{j=0}^{L} c_j A^j$. The circuit uses at most 3 ancilla qubits, $\mathcal{O}(L)$ queries to $U$ and $U^\dagger$, $\mathcal{O}(1)$ queries to controlled-$U$, and $\mathcal{O}((m+1)L)$ single- and two-qubit gates.