
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 point $P$ lies on the ellipse $\frac{x^2}{9} + \frac{y^2}{5} = 1$, and point $P$ is not on the $x$-axis. Points $A$ and $B$ are the left and right vertices of the ellipse. Line $PA$ intersects the $y$-axis at point $C$. The slopes of lines $BC$ and $PB$ are $k_{BC}$ and $k_{PB}$ respectively. Find the minimum value of ${k_{BC}}^2 + 2{k_{PB}}^2$.

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