
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
The graph of \(y = f(x)\) is shown below, composed of three pieces:
1. \(f(x) = -2 - x\) for \(-3 \leq x \leq 0\),
2. \(f(x) = \sqrt{4 - (x - 2)^2} - 2\) for \(0 \leq x \leq 2\),
3. \(f(x) = 2(x - 2)\) for \(2 \leq x \leq 3\).

Define the function \(y = h(x)\) by modifying \(f(x)\) as follows:
1. Double the output of \(f(x)\) for the linear piece,
2. Keep the circular segment unchanged,
3. For the linear ascending segment, add 2 units to the output of \(f(x)\).

Express \(h(x)\) in terms of \(f(x)\).

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