
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 $P$ moves on $y=\sqrt{{x}^{2}+1},x∈[-1,\sqrt{3}]$, and point $Q$ moves on the circle $C:{x}^{2}+(y-a)^{2}=\frac{3}{4}(a＞0)$, and $|PQ|$ has a minimum value of $\frac{3}{2}\sqrt{3}$, then the value of the real number $a$ is ______.

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