
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 \( F_1 = \left( -2, 2 - \frac{\sqrt{3}}{2} \right) \) and \( F_2 = \left( -2, 2 + \frac{\sqrt{3}}{2} \right) \). The set of points \( P \) such that
\[
|PF_1 - PF_2| = \sqrt{2}
\]
forms a hyperbola. The equation of this hyperbola can be written as
\[
\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1,
\]
where \( a, b > 0 \). Find \( h + k + a + b \).

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