
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
Let $P$ be any point on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\ (a>b>0)$ with foci $F_{1}(-c, 0)$ and $F_{2}(c, 0)$. The lines $P F_{1}$ and $P F_{2}$ intersect the ellipse at points $A$ and $B$ respectively. If $a^{2}$, $b^{2}$, and $c^{2}$ form an arithmetic sequence, then find the value of $\frac{\left|P F_{1}\right|}{\left|A F_{1}\right|}+\frac{\left|P F_{2}\right|}{\left|B F_{2}\right|}$.

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