
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 hyperbola \({{x}^{2}}-\dfrac{{{y}^{2}}}{{{b}^{2}}}=1\) has an eccentricity of \(\dfrac{\sqrt{5}}{2}\), with the left focus being \({{F}_{1}}\). When point \(P\) moves on the right branch of the hyperbola and point \(Q\) moves on the circle \({{x}^{2}}+{{\left( y-1 \right)}^{2}}=1\), the minimum value of \(\left| PQ \right|+\left| PF_{1} \right|\) is \_\_\_\_\_.

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