
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 point $P_{i}(x_{i},y_{i})$ on line $l_{i}$: $a_{i}x+b_{i}y=c_{i}$, if $a_{i}+b_{i}=ic_{i}(i=1,2)$, and $|P_{1}P_{2}|\geqslant \frac{\sqrt{2}}{2}$ always holds true, then what is the value of $\frac{c_{1}}{a_{1}}+\frac{a_{2}}{c_{2}}$?

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