
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 the moving straight line ${l_{0}}:ax+by+c-2=0 (a > 0, c > 0)$ that always passes through point $P(1,m)$, and the maximum distance from point $Q(4,0)$ to the moving straight line ${l_{0}}$ is $3$, find the minimum value of $\frac{1}{2a}+\frac{2}{c}$.

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