
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

...

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system
Let \(\xi = (\xi_k, \mathscr{D}_k)_{0 \leqslant k \leqslant n}\) be a submartingale,

\[ 
\xi_k = \sum_{m \leqslant k} I_{A_m}, 
\]

where \(A_m \in \mathscr{D}_m\). Provide the Doob decomposition for this submartingale.

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