
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
In the Cartesian coordinate plane $xOy$, two circles $C_1$ and $C_2$ are defined as follows: $C_1: x^2+(y-t)^2=4$ and $C_2: (x-2)^2+y^2=14$. If there exists a point $P$ on circle $C_1$ such that a tangent line to circle $C_2$ passes through $P$ at point $Q$ and $PO = \sqrt{2}PQ$, determine the range of values for the real number $t$.

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