
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 4 lines in Euclidean 3-space:
\[ L_1 : x = 1, y = 0 \]
\[ L_2 : y = 1, z = 0 \]
\[ L_3 : x = 0, z = 1 \]
\[ L_4 : x = y, y = -6z \]

Find the equations of the two lines which both meet all of the \( L_i \) (for \( i = 1, 2, 3, 4 \)).

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