
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 a point $P(x,y)$ moves on the line $x+2y=3$, when $2^{x}+4^{y}$ takes the minimum value, a tangent line is drawn from point $P$ to the circle $\left(x-\frac{1}{2}\right)^{2}+\left(y+\frac{1}{4}\right)^{2}=\frac{1}{2}$. The length of this tangent segment is __________.

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