
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 triangle $ABC$, $\angle BAC = 45^\circ$ and $BC = 1$. Triangle $ABC$ is inscribed in a circle with center $O$, and incircle with center $I$ touches $AB$ at point $D$. Assume $\angle DBC = 15^\circ$. Let $H$ be the orthocenter of $\triangle ABC$. Find the length of segment $HI$.
**A)** \( \frac{1}{2} \) \\
**B)** \( \sqrt{2} - 1 \) \\
**C)** \( \sqrt{2} - \frac{1}{2} \) \\
**D)** \( 1 \)

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