= Solution
Fix the sign convention $\iota_{\xi_M}\omega=-d\mu^\xi$, where $\xi_M$ is the <fundamental vector field> and $\mu^\xi=\langle\mu,\xi\rangle$. A <Hamiltonian group action> is a symplectic <Lie group action> together with an equivariant <moment map> $\mu:M\to\mathfrak g^*$ satisfying this identity for every $\xi\in\mathfrak g$. Equivariance means $\mu(gx)=\operatorname{Ad}_{g^{-1}}^*\mu(x)$. Merely requiring each fundamental field to have some Hamiltonian is the weaker condition of a <weakly Hamiltonian action>.
For a <circle group> action, identify its Lie algebra with $\mathbb R$, and let $Y$ generate the action. Put $Z=\mu^{-1}(t)$ and let $i:Z\hookrightarrow M$. Since $t$ is a <regular value>,
$$
T_xZ=\ker d\mu_x=\{v:\omega(Y_x,v)=0\}.
$$
The action is free, so $Y_x\ne0$. The <symplectic orthogonal complement> of this hyperplane is exactly $\mathbb RY_x$, which is contained in $T_xZ$. Thus
$$
\ker(i^*\omega)_x=\mathbb RY_x.
$$
The free action of the compact <circle group> is proper, and the quotient $Q=Z/S^1$ is a <smooth manifold> with projection $\pi:Z\to Q$ a <principal bundle>. The form $i^*\omega$ is invariant and kills the orbit directions, hence is basic and descends to a unique two-form $\omega_Q$ satisfying $\pi^*\omega_Q=i^*\omega$. It is closed because pullback by a <submersion> detects <differential forms>. If a quotient tangent vector annihilates $\omega_Q$, a lift lies in the displayed <kernel> and hence projects to zero. Therefore $\omega_Q$ is nondegenerate. This proves <symplectic reduction> directly.
If $L\subset Q$ is a <Lagrangian submanifold>, its inverse image $\widetilde L=\pi^{-1}(L)$ is a smooth <circle bundle> over $L$. The restricted <symplectic form> is $\pi^*(\omega_Q|_L)=0$. If $\dim M=2N$, then $\dim Q=2N-2$, $\dim L=N-1$, and $\dim\widetilde L=N$, proving that $\widetilde L$ is Lagrangian. This is the <Lagrangian lift through circle reduction>.
For scalar multiplication on $\mathbb C^{n+1}$, the generator is $Y=\sum_j(-y_j\partial_{x_j}+x_j\partial_{y_j})$. Direct contraction gives
$$
\iota_Y\omega_{st}=-\sum_j(x_jdx_j+y_jdy_j),\qquad\boxed{\mu(z)=\tfrac12|z|^2.}
$$
Choose $t>0$. Its level is the sphere of radius $R=\sqrt{2t}$ and its quotient is $\mathbb{CP}^n$. <Complex conjugation> on the sphere reverses the ambient <symplectic form> and descends to an <anti-symplectic involution> of the quotient. Its fixed set is $\mathbb{RP}^n$: a projective point fixed by conjugation has a real representative after a phase change. For two tangent vectors to this fixed set, anti-symplecticity makes the reduced form equal to its negative, so it vanishes. The real dimension is $n$, half the quotient dimension, proving the <Lagrangian fixed locus of an anti-symplectic involution> property.
The lifted <Lagrangian submanifold> can be written explicitly as
$$
\boxed{\widetilde L=\{e^{i\theta}x:x\in\mathbb R^{n+1},\ |x|=R\}
\cong(S^n\times S^1)/((x,\zeta)\sim(-x,-\zeta)).}
$$
The only redundancy in this parametrization is the displayed free involution: two nonzero real representatives on the same complex ray differ by a real sign. The quotient embeds smoothly in the sphere, and projection to $[x]\in\mathbb{RP}^n$ exhibits its <circle bundle> structure. Its Lagrangian property follows from the reduction argument, including the case $n=0$.
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