
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 that points $M$ and $N$ are moving points on the parabola $C: y^{2}=4x$ and the circle $D: (x+4)^{2}+y^{2}=4$ respectively. The distance from point $M$ to the directrix of the parabola $C$ is $d$. Find the minimum value of $|MN|+d$.

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