###### Figure 3

(a) For some

$q(\theta )$, the self-consistent equation

$R=S[R,K,q(\theta )]$ has two solutions,

${r}_{1}<{r}_{2}$ (brown dashed line). Setting

$\alpha $ appropriately makes

$R=S[R,K,\alpha q(\theta )]$ have only one solution

$R=r<{r}_{2}$ (red solid line). Note that

${S}^{\prime}[r,K,\alpha q(\theta )]=1$. (b) If we have

$S[r,K,{q}_{1}(\theta )]=r$ and

${S}^{\prime}[r,K,{q}_{1}(\theta )]={s}_{1}\le 1$ (blue short-dashed line) and

$S[r,K,{q}_{2}(\theta )]=r$ and

${S}^{\prime}[r,K,{q}_{2}(\theta )]={s}_{2}\ge 1$ (green long-dashed line), we can obtain

$q(\theta )$ with which

$S[r,K,q(\theta )]=r$ and

${S}^{\prime}[r,K,q(\theta )]=1$ hold (red solid line). (c,d,e)

${q}_{2}(\theta )$ which maximizes

${S}^{\prime}[r,K,q(\theta )]$ (e) under the constraint

$S[r,K,q(\theta )]=r$ (d) is given by

${q}_{2}(\theta )=2\Theta [W(\theta ,r,K)-{w}_{2}]-1$ (green long-dashed line) where

${w}_{2}$ is set to satisfy

$S[r,K,{q}_{2}(\theta )]=r$ (c). Under the same constraint,

${S}^{\prime}[r,K,q(\theta )]$ is minimized by

${q}_{1}(\theta )=2\Theta [{w}_{1}-W(\theta ,r,K)]-1$ (blue short-dashed line) where

$S[r,K,{q}_{1}(\theta )]=r$. (f) Attainable region of the order parameter

$R$ (shaded orange region). Note that the incoherent state

$R=0$ is stable for any

$K$ (red solid line). (g) Phase diagram of the system of phase oscillators when the strength of two-body and three-body interactions are changed. The symbol in each region is a schematic representation of the attainable values of the order parameter

$R$. Gray lines represent the range of

$R$ from 0 to 1. The attainable values and ranges of

$R$ are indicated by black circles and boxes, respectively.

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