
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
Let point $P_i(x_i, y_i)$ be on the line $l_i: a_ix + b_iy = c_i$, if $a_i + b_i = ic_i$ (where $i=1,2$), and $|P_1P_2| \geq \frac{\sqrt{2}}{2}$ always holds, then $\frac{c_1}{a_1} + \frac{a_2}{c_2} =$ \_\_\_\_\_\_.

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