
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 an isosceles triangle \(ABC\) with base \(AC\), the side \(AB\) is equal to 2, segment \(AD\) is the angle bisector. A tangent \(DH\) to the circumcircle of triangle \(ADB\) passes through point \(D\) and point \(H\) lies on side \(AC\). Find the area of triangle \(ABC\) if \(CD = \sqrt{2} \cdot CH\).

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