
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 points $F_1(-c,0)$ and $F_2(c,0)$ ($c > 0$) are the left and right foci of the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), point $P$ is a point on this ellipse located above the $x$-axis, and point $G$ is the circumcenter of $\triangle PF_1F_2$. If there exists a real number $\lambda$ such that $\overrightarrow{GF_1} + \overrightarrow{GF_2} + \lambda \overrightarrow{GP} = \overrightarrow{0}$, then when the area of $\triangle PF_1F_2$ is $8$, the minimum value of $a$ is ______.

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