
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 edge of the cube \(E F G H E_1 F_1 G_1 H_1\) is equal to 2. Points \(A\) and \(B\) are taken on the edges \(E H\) and \(H H_1\) such that \(\frac{E A}{A H} = 2\) and \(\frac{B H}{B H_1} = \frac{1}{2}\). A plane is drawn through the points \(A\), \(B\), and \(G_1\). Find the distance from point \(E\) to this plane.

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