
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
$(1)$ Given $a$, $b$, $c$, $d \in \mathbb{R}$, compare $(a^{2}+b^{2})(c^{2}+d^{2})$ with $\left(ac+bd\right)^{2}$ and explain the reason.<br/>$(2)$ Use the conclusion of (1) to solve the following problem: Given $a$, $b$ are positive numbers, and $a^{2}+b^{2}=5$, find the maximum value of $2a+b$.

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