
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 $P(x,y)$ moves on the line $x+2y=3$, when $2^{x}+4^{y}$ reaches its minimum value, find the length of the tangent line drawn through point $P(x,y)$ to the circle $C: (x-\frac{1}{2})^{2}+(y+\frac{1}{4})^{2}=\frac{1}{2}$.

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