
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 $P_{1}P_{2}$ is a line segment on the line $\begin{cases} x=1+\frac{1}{2}t \\ y=-2+\frac{\sqrt{3}}{2}t \end{cases}$ (where $t$ is a parameter), and the parameters corresponding to points $P_{1}$ and $P_{2}$ are $t_{1}$ and $t_{2}$ respectively, find the distance from the midpoint $P$ of the line segment $P_{1}P_{2}$ to the point $(1,-2)$.

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