
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 the hyperbola $C$: $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$, its right vertex is $A$, and a circle $A$ with center $A$ and radius $b$ intersects one of the asymptotes of the hyperbola $C$ at points $M$ and $N$. If $\angle MAN = 60^{\circ}$, then the eccentricity of $C$ is ______.

(2) The equation of one of the asymptotes of the hyperbola $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{9} = 1$ $(a > 0)$ is $y = \dfrac{3}{5}x$, then $a=$ ______.

(3) A tangent line to the circle $x^{2} + y^{2} = \dfrac{1}{4}a^{2}$ passing through the left focus $F$ of the hyperbola $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$ intersects the right branch of the hyperbola at point $P$. If $\overrightarrow{OE} = \dfrac{1}{2}(\overrightarrow{OF} + \overrightarrow{OP})$, then the eccentricity of the hyperbola is ______.

(4) A line passing through the focus $F$ of the parabola $y^{2} = 2px$ $(p > 0)$ with an inclination angle of $\dfrac{\pi}{4}$ intersects the parabola at points $A$ and $B$. If the perpendicular bisector of chord $AB$ passes through point $(0,2)$, then $p=$ ______.

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