
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 that $F_{1}$ and $F_{2}$ are two foci of the hyperbola $\frac{x^2}{4}-\frac{y^2}{b^2}=1(b＞0)$, point $A$ is the right vertex of the hyperbola, and $M(x_{0}$,$y_{0})(x_{0} \gt 0$,$y_{0} \gt 0)$ is a point on the asymptote of the hyperbola, satisfying $MF_{1}\bot MF_{2}$. If the parabola with focus at $A$ is $y^{2}=2px\left(p \gt 0\right)$ passing through point $M$, then the eccentricity of this hyperbola is ______.

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