JavaScript (Node.js), 95 bytes

a=>(N=g=([p,q,...x],n,...s)=>1/p?g(x,n+p,...s,p)+g([q,...x],n,...s):N>n|s<g||[N=n,S=s])(a,0)&&S

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awful

JavaScript (Node.js), 91 bytes

g=([p,q,...x],n=N=-1/0,...s)=>1/p?g(x,~~n+p,...s,p)&&g([q,...x],n,...s):N>n|s<g?S:(N=n,S=s)

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JavaScript (Node.js), 95 94 bytes

a=>(N=g=([p,q,...x],n,...s)=>1/p?g(x,n+p,...s,p)&&g([q,...x],n,...s):N>n|s<g?S:(N=n,S=s))(a,0)

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