[10%] You are a process engineer managing a critical Lithography Machine. The machine has a strict Thermal Constraint:

Constraint: You cannot schedule production for two consecutive shifts. If Shift $i$ is active, Shift $i-1$ and Shift $i+1$ must be idle (cooling).

Data: The estimated Production Yield Value for the upcoming 13 shifts is listed below.

Questions:

(1) [4%] Identify the set of shifts that results in the maximum possible total value while satisfying the thermal constraint. (Selected Shifts、Total Value)

(2) [4%] A very important client demands that Shift 7 (Value: 40) MUST be included in the schedule. Under this mandatory condition, what is the maximum total value you can achieve for the 13 shifts? (Selected Shifts with S7、Total Value with S7)

(3) [2%] Process Upgrade: The High-Temperature Machine. Imagine the factory installs a new, high-performance machine. This machine generates significantly more heat, so the safety constraint is updated:

New Constraint: After any active shift, the machine must cool down for TWO consecutive shifts.

Let $f[i]$ be the maximum value achievable considering shifts up to $i$, and let $v_i$ be the value of Shift $i$. Please write down the complete Recurrence Relation (Mathematical Equation) for $f[i]$.