
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Given a polar coordinate system with the origin \(O\) as the pole and the non-negative half of the \(x\)-axis as the polar axis, where the length units in both coordinate systems are the same, the polar equation of curve \(C_1\) is \(\rho\sin^2\theta - 4\cos\theta = 0\), and the parametric equation of curve \(C_2\) is \[ \begin{cases} x=-1+2\cos\phi \\ y=2\sin\phi \end{cases} \] where \(\phi\) is the parameter.
(1) Find the Cartesian equation of \(C_1\) and the general equation of \(C_2\);
(2) Given the point \(P\left(\frac{1}{2},0\right)\), and the parametric equation of line \(l\) is \[ \begin{cases} x=\frac{1}{2}+\frac{\sqrt{2}}{2}t \\ y=\frac{\sqrt{2}}{2}t \end{cases} \] where \(t\) is the parameter. If line \(l\) intersects curve \(C_1\) at points \(M\) and \(N\), find the value of \(\frac{1}{|PM|} + \frac{1}{|PN|}\).

Present the answer in LaTex format: \boxed{Your answer}
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