
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 \( F_1 = (1, 2) \) and \( F_2 = (5, 2) \). The set of points \( P \) such that
\[ PF_1 + PF_2 = 10 \]
forms an ellipse. The equation of this ellipse can be written as
\[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1. \]
Find \( h + k + a + b \).

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