
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) Write the equation of the parabola with vertex $M$.

(2) Let $N$ and $F$ be points on the $x$-axis and $y$-axis, respectively. When the perimeter of the quadrilateral with vertices $M$, $D$, $N$, and $F$ is minimized, find the coordinates of points $N$ and $F$, and determine the minimum perimeter.

(3) A moving point $P$ starts from point $O$ and moves along the direction of $OC$ at a speed of 1 unit length per second. Given the time of movement as $t$ seconds, for what value of $t$ is the angle $\angle BPD$ maximized?

Find all four-digit numbers $m$ such that $m \leq 2025$ and there exists a positive integer $n$ such that $m - n$ is a prime number and $mn$ is a perfect square.

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