
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 plane vectors $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}$ satisfy $|\boldsymbol{a}|=1, \boldsymbol{a} \cdot \boldsymbol{b}=\boldsymbol{b} \cdot \boldsymbol{c}=1,$ and $|\boldsymbol{a}-\boldsymbol{b}+\boldsymbol{c}| \leqslant 2 \sqrt{2}$. Find the maximum value of $\boldsymbol{a} \cdot \boldsymbol{c}$.

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