
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
Triangle ABC has $\angle BAC = 45^\circ$, $\angle CBA \leq 120^\circ$, $BC=2$, and $AC = 2 \cdot AB$. Let H, I, and O be the orthocenter, incenter, and circumcenter of $\triangle ABC$, respectively. Find the measure of ∠CBA if the area of pentagon BCOIH is the maximum possible.

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