Polynomial transformation on block encodings (Hermitian, real with parity)

Let $m,n,L\geq0$ be integers, $P=\sum_{j=0}^{L}c_jx^j$ be a degree-$L$ real polynomial with parity $L\bmod2$ such that $|P(x)|\leq1$ for every $x\in[-1,1]$, and $U$ be a $(1,m,0)$-block-encoding of an $n$-qubit Hermitian matrix $A$. Given access to $U$ and $U^\dagger$, there exists a quantum circuit $\mathcal V$ that implements a $(1,m+1,0)$-block-encoding of $P(A)=\sum_{j=0}^{L}c_jA^j$, using at most 1 ancilla qubit, $L$ queries to $U$ and $U^\dagger$, $192mL$ CNOT gates, and $((296m+3)L+2)$ single-qubit gates.