
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
(1) Given a point $P$ on the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1$ whose distance to one focus of the ellipse is $3$, the distance from $P$ to the other focus is ______.

(2) The equation of the asymptotes of the hyperbola $\dfrac{x^2}{16} - \dfrac{y^2}{64} = 1$ is ________.

(3) Given that the equation of the tangent line to the graph of the function $y=f(x)$ at the point $M(1, f(1))$ is $y= \dfrac{1}{2}x+2$, then $f(1)+f'(1)=$ ______.

(4) Given that the function $f(x)=2a\ln x + x^2 - 2x$ $(a \in \mathbb{R})$ is monotonically increasing over its domain, the minimum value of $a$ is ______.

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