
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 the parabola $C: x^2 = 2py$ ($p > 0$), the distance from the focus $F$ to the directrix is $\frac{1}{2}$.
(1) Determine the equation of the parabola $C$;
(2) Let point $P$ on parabola $C$ have an $x$-coordinate of $t$ ($t > 0$). A line passing through $P$ intersects $C$ again at point $Q$ and the $x$-axis at point $M$. A perpendicular to $PQ$ through point $Q$ intersects $C$ at another point $N$. If $MN$ is a tangent to $C$, find the minimum value of $t$.

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