
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
Take the origin \\(O\\) of the Cartesian coordinate system as the pole, the positive half-axis of the \\(x\\)-axis as the polar axis, and both coordinate systems have equal unit lengths. The parametric equation of line \\(l\\) is \\(\begin{cases} & x=\dfrac{\sqrt{2}}{2}t, \\\\ & y=3+\dfrac{\sqrt{2}}{2}t \end{cases}\\) (where \\(t\\) is the parameter), and the polar equation of curve \\(C\\) is \\(\rho \cos^2\theta =2\sin \theta\\).

\\((1)\\) Write the standard equation of line \\(l\\) and the Cartesian coordinate equation of curve \\(C\\);

\\((2)\\) Suppose line \\(l\\) intersects curve \\(C\\) at points \\(A\\) and \\(B\\), \\(M\\) is the midpoint of \\(AB\\), and point \\(P\\) has polar coordinates \\((\sqrt{2},\dfrac{\pi}{4})\\), find the value of \\(|PM|\\).

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