
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 the focus of the parabola $C: y^{2}=4x$ is $F$, two lines $l_{1}$ and $l_{2}$ are drawn passing through point $F$. Line $l_{1}$ intersects the parabola $C$ at points $A$ and $B$, while line $l_{2}$ intersects the parabola $C$ at points $M$ and $N$. If the product of the slopes of $l_{1}$ and $l_{2}$ is $-1$, calculate the minimum value of $|AB|+|MN|$.

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