
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
Given that the function $f(x)$ is an even function defined on $\mathbb{R}$ and for any $x \in \mathbb{R}$ it satisfies $f(x+2)=f(x)$, with $f(x)=x^2$ for $0 \leq x \leq 1$, find the value of the real number $a$ such that the line $y=x+a$ has two distinct common points with the graph of $y=f(x)$.

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