
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 \(a_i\) and \(b_i\) be real numbers such that \(a_1 b_2 \neq a_2 b_1\). What is the maximum number of possible 4-tuples \((\text{sign}(x_1), \text{sign}(x_2), \text{sign}(x_3), \text{sign}(x_4))\) for which all \(x_i\) are non-zero and \(x_i\) is a simultaneous solution of
\[
a_1 x_1 + a_2 x_2 + a_3 x_3 + a_4 x_4 = 0
\]
and
\[
b_1 x_1 + b_2 x_2 + b_3 x_3 + b_4 x_4 = 0.
\]
Find necessary and sufficient conditions on \(a_i\) and \(b_i\) for this maximum to be achieved.

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