
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 $F_{1}(-c,0)$, $F_{2}(c,0)$ $(c > 0)$ are the left and right foci of the ellipse $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$ $(a > b > 0)$, and point $P$ is a point on this ellipse located above the $x$-axis. Point $G$ is the circumcenter of $\triangle PF_{1}F_{2}$. Find the minimum value of $a$ when the area of $\triangle PF_{1}F_{2}$ is $8$.

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