
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 hyperbola $C$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$) with a focal length of $8$. A line passing through the left focus $F$ intersects the left branch of $C$ at points $A$ and $B$. Perpendiculars are drawn from $A$ and $B$ to the line $l: x = -1$, with the feet of the perpendiculars denoted as $M$ and $N$ respectively. When $AB$ is perpendicular to the $x$-axis, $|MN| = 12$.
1. Find the standard equation of $C$.
2. Let $P(2\sqrt{3}-1, 0)$. Determine if there exists $t > 0$ such that $\frac{1}{|PM| - t} + \frac{1}{|PN| - t}$ is a constant value. If it exists, find the value of $t$; if not, explain the reason.

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