
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
In the rectangular coordinate system on a plane, the center of the hyperbola Γ is at the origin, one of its focal points has coordinates $( \sqrt {5} , 0 )$, and $\overrightarrow {e}_{1} = (2, 1)$ and $\overrightarrow {e}_{2} = (2, -1)$ are the direction vectors of the two asymptotes, respectively. For any point P on the hyperbola Γ, if $\overrightarrow {OP} = \overrightarrow {ae}_{1} + \overrightarrow {be_{2}}$ (where $a, b \in \mathbb{R}$), then the equation that $a$ and $b$ satisfy is ________.

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