
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 the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{3}=1\) with the left vertex \(A_{1}\) and the right focus \(F_{2}\), let \(P\) be a moving point on the ellipse. When \(\overrightarrow{PA_{1}} \cdot \overrightarrow{PF_{2}}\) takes its minimum value, calculate the value of \(\left|\overrightarrow{PA_{1}}+\overrightarrow{PF_{2}}\right|\).

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