
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
Let the sequence \(\left\{a_{n}\right\}\) satisfy the condition that for all positive integers \(n\),
\[ 1+\sum_{m=1}^{n} \mathrm{C}_{n}^{m} a_{m}=2^{\frac{n}{2}} \cos \frac{n \pi}{4}. \]

Find the general formula for the sequence \(\left\{a_{n}\right\}\).

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