
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
(1) Given that point $M$ lies on the parabola $y^{2}=4x$, point $F$ is the focus of the parabola, and point $P(3,1)$ is fixed, find the minimum value of $MP+MF$.

(2) Given the parabola $y^{2}=2x$, let $A(a,0)$ where $a > 0$, and point $P$ lies on the parabola such that $PA=d$, find the minimum value of $d$.

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