
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
Let $e$ be the tangent to the hyperbola $y = \frac{1}{x}$ at the point $P$ with abscissa $a \geq 1$. Denote by $t(a)$ the area enclosed by the tangent $e$ and the coordinate axes, and by $T(a)$ the area of the triangle bounded by the tangent $e$, the perpendicular to $e$ at $P$, and the x-axis. Does the ratio $\frac{t(a)}{T(a)}$ have a maximum and minimum value?

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