
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 points $M$ and $N$ are two different points on the parabola $y=4x^{2}$, and $F$ is the focus of the parabola, satisfying $\angle MFN=135^{\circ}$. The midpoint of chord $MN$ is $P$, and the distance from $P$ to the line $l: y=-\frac{1}{16}$ is denoted as $d$. If $|MN|^{2}=\lambda \cdot d^{2}$, then the minimum value of $\lambda$ is \_\_\_\_\_\_.

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